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update to LOOPS 3.4.0

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These are simply the changes as distributed.
This commit is contained in:
Glen Whitney 2017-10-29 23:54:13 -04:00
parent 7e8b3b5562
commit f64208f12f
58 changed files with 17724 additions and 29097 deletions
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8
PackageInfo.g
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@ -1,9 +1,9 @@
SetPackageInfo( rec( SetPackageInfo( rec(
PackageName := "loops", PackageName := "loops",
Subtitle := "Computing with quasigroups and loops in GAP", Subtitle := "Computing with quasigroups and loops in GAP",
Version := "3.3.0", Version := "3.4.0",
Date := "26/10/2016", Date := "27/10/2017",
ArchiveURL := "http://www.math.du.edu/loops/loops-3.3.0", ArchiveURL := "http://www.math.du.edu/loops/loops-3.4.0",
ArchiveFormats := "-win.zip .tar.gz", ArchiveFormats := "-win.zip .tar.gz",
Persons := [ Persons := [
@ -83,7 +83,7 @@ Dependencies := rec(
), ),
AvailabilityTest := ReturnTrue, AvailabilityTest := ReturnTrue,
BannerString := "This version of LOOPS is ready for GAP 4.7.\n", BannerString := "This version of LOOPS is ready for GAP 4.8.\n",
Autoload := false, # false for deposited packages Autoload := false, # false for deposited packages
TestFile := "tst/testall.g", TestFile := "tst/testall.g",

41
data/automorphic.tbl
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@ -2,39 +2,21 @@
## ##
#W automorphic.tbl Automorphic loops G. P. Nagy / P. Vojtechovsky #W automorphic.tbl Automorphic loops G. P. Nagy / P. Vojtechovsky
## ##
#H @(#)$Id: automorphic.tbl, v 3.3.0 2016/10/20 gap Exp $ #H @(#)$Id: automorphic.tbl, v 3.4.0 2017/10/23 gap Exp $
## ##
#Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary), #Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary),
#Y P. Vojtechovsky (University of Denver, USA) #Y P. Vojtechovsky (University of Denver, USA)
## ##
#############################################################################
## Binding global variables
## LOOPS_automorphic_cocycles
## LOOPS_automorphic_bases
## LOOPS_automorphic_coordinates
# Many small automorphic loops are represtented by encoded Cayley tables.
#
# Commutative automorphic loops of order 243 are represtented as central
# extensions of the cyclic group of order 3.
# The necessary data is only loaded on demand and consists of:
# - LOOPS_automorphic_cocycles, a list of encoded bases of the
# space of cocycles modulo coboundaries for every factor loop F needed.
# - LOOPS_automorphic_coordinates, a list that for every loop
# points to the factor loop and gives coordinates of the required cocycle
# with respect to the relevant basis.
LOOPS_automorphic_data := [ LOOPS_automorphic_data := [
#implemented orders #implemented orders
[3,6,8,9,10,12,14,15,27,81,243], [3,6,8,9,10,12,14,15,27,81],
#number of nonassociative loops of given order #number of nonassociative loops of given order
[1,1,7,2,3,2,5,2,7,72,118451], [1,1,7,2,3,2,5,2,7,72],
#the loops #the loops
[ [
#order 3 (Z_3) #order 3 (Z_3, use left Bruck loops, placeholder only)
[ [
"201"
], ],
#order 6 #order 6
[ [
@ -50,10 +32,8 @@ LOOPS_automorphic_data := [
"0325476301675421076455760132467102374523106543201", "0325476301675421076455760132467102374523106543201",
"0325476310674520176545761023467013275432106452301" "0325476310674520176545761023467013275432106452301"
], ],
#order 9 (two abelian groups) #order 9 (two abelian groups, use left Bruck loops, placeholder only)
[ [
"204537861534867678012861207201345534",
"204537861534867678120862017012453345"
] ]
, ,
#order 10 #order 10
@ -80,20 +60,13 @@ LOOPS_automorphic_data := [
"234068597BDAEC340189675DEBCA401297856ECDAB012375968CAEBD6897ADECB041328975DCABE430215689EABDC102439756CBDEA324107568BECAD21304BDEC0413258976DECA4302187569ABDE1024395687ECAB3241076895CABD2130469758", "234068597BDAEC340189675DEBCA401297856ECDAB012375968CAEBD6897ADECB041328975DCABE430215689EABDC102439756CBDEA324107568BECAD21304BDEC0413258976DECA4302187569ABDE1024395687ECAB3241076895CABD2130469758",
"234067895BCDEA340178956CDEAB401289567DEABC012395678EABCD7968ADBEC012348579ECADB340129685DBECA123405796CADBE401236857BECAD23401DBEC0432156789ECAD3210478956ADBE1043295678BECA4321067895CADB2104389567" "234067895BCDEA340178956CDEAB401289567DEABC012395678EABCD7968ADBEC012348579ECADB340129685DBECA123405796CADBE401236857BECAD23401DBEC0432156789ECAD3210478956ADBE1043295678BECA4321067895CADB2104389567"
], ],
#order 27 (commutative only, placeholder) #order 27 (commutative only, use left Bruck loops, placeholder)
[ [
] ]
, ,
#order 81 (commutative only, placeholder) #order 81 (commutative only, use left Bruck loops, placeholder)
[
]
,
#order 243 (commutative only, placeholder)
[ [
] ]
] ]
]; ];
LOOPS_automorphic_cocycles := [];
LOOPS_automorphic_bases := [];
LOOPS_automorphic_coordinates := [];

8058
data/automorphic/automorphic_cocycles.tbl
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61
data/cc.tbl
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@ -1,31 +1,52 @@
############################################################################# #############################################################################
## ##
#W cc.tbl CC-loops p^2, 2p, for p odd prime G. P. Nagy / P. Vojtechovsky #W cc.tbl Library of CC loops G. P. Nagy / P. Vojtechovsky
## ##
#H @(#)$Id: cc.tbl, v 3.0.0 2015/06/10 gap Exp $ #H @(#)$Id: cc.tbl, v 3.4.0 2015/06/10 gap Exp $
## ##
#Y Copyright (C) 2005, G. P. Nagy (University of Szeged, Hungary), #Y Copyright (C) 2005, G. P. Nagy (University of Szeged, Hungary),
#Y P. Vojtechovsky (University of Denver, USA) #Y P. Vojtechovsky (University of Denver, USA)
## ##
#############################################################################
## Binding global variables
## LOOPS_cc_used_factors
## LOOPS_cc_cocycles
## LOOPS_cc_bases
## LOOPS_cc_coordinates
# CC loops are activated as follows: # CC loops are activated as follows:
# If n = 2p or p^2, where p is a prime, then we call a method for # If n = 2p, where p is an odd prime, then we call an algebraic method for
# cosntructing these loops. # constructing these loops.
# If n = p^2, where p>3 is a prime, then we call an algebraic method for
# construction these loops.
# If n is a power of 2 or 3, then we use cocycles located in cc/cc_cocycles_n.tbl.
# For all other orders, we point to the library of RCC loops. # For all other orders, we point to the library of RCC loops.
LOOPS_cc_data := [ LOOPS_cc_data := [
#implemented orders #implemented orders
[ 8, 12, 16, 18, 20, 21, 24, 27], [ 2, 3, 4, 5, 7, 8, 9, 12, 16, 18, 20, 21, 24, 25, 27, 32, 49, 64, 81, 125, 343],
#number of nonassociative loops of given order #number of loops of given order in the library
[ 2, 3, 28, 7, 3, 1, 14, 55], [ 1, 1, 2, 1, 1, 7, 5, 3, 42, 7, 3, 1, 14, 5, 60, 437, 5, 14854, 5406, 84, 122],
#the numbers of the loops in the RCC library
[ [
#order 8 #order 2 (Z_2)
[2,7], ["010"],
#order 3 (Z_3)
["201"],
#order 4 (placeholder only)
,
# order 5 (Z_5)
["2340401123"],
# order 7 (Z_7)
["234560456016012123345"],
#order 8 (placeholder only)
,
#order 9 (placeholder only)
,
#order 12 #order 12
[53,73,89], [53,73,89],
#order 16 #order 16 (placeholder only)
[9,35,107,228,243,292,437,440,1043,1883,1936,2332,2420,2636,2645,2750,2753,2794,2797,2847,3682,3730,3739,3848,3949,4735,4904,4925], ,
#order 18 #order 18
[22,29,77,292,360,377,1133], [22,29,77,292,360,377,1133],
#order 20 #order 20
@ -33,18 +54,12 @@ LOOPS_cc_data := [
#order 21 #order 21
[104], [104],
#order 24 #order 24
[302,1025,2119,2182,2335,3066,4569,5176,5589,5997,7495,194830,225705,243216], [302,1025,2119,2182,2335,3066,4569,5176,5589,5997,7495,194830,225705,243216]
#order 27
[78,86,317,319,361,571,711,1080,1085,1624,1665,2217,2219,3614,3624,8579,8582,15059,15072,15503,15512,19439,23177,23214,26331,26348,52978,55027,55055,59116,59123,75864,78970,79011,83042,83104,83155,104913,106081,106144,110854,110892,110930,114102,117212,119407,134858,136370,140791,148160,148892,149330,151792,152090,152515]
] ]
]; ];
# The following can be used to point to CC loops of order 2p and p^2 in the library of RCC loops. LOOPS_cc_used_factors := [];
# order 6, [3] LOOPS_cc_cocycles := [];
# order 9, [5,4,3] LOOPS_cc_bases := [];
# order 10, [16] LOOPS_cc_coordinates := [];
# order 14, [97]
# order 22, [10346]
# order 25, [86,93,118]
# order 26, [151964]

61
doc/chap0.txt
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@ -6,7 +6,7 @@
Computing with quasigroups and loops in GAP Computing with quasigroups and loops in GAP
Version 3.3.0 Version 3.4.0
Gábor P. Nagy Gábor P. Nagy
@ -28,7 +28,7 @@
------------------------------------------------------- -------------------------------------------------------
Copyright Copyright
© 2016 Gábor P. Nagy and Petr Vojtěchovský. © 2017 Gábor P. Nagy and Petr Vojtěchovský.
------------------------------------------------------- -------------------------------------------------------
@ -167,10 +167,12 @@
6.11-3 QuasigroupsUpToIsomorphism 6.11-3 QuasigroupsUpToIsomorphism
6.11-4 LoopsUpToIsomorphism 6.11-4 LoopsUpToIsomorphism
6.11-5 AutomorphismGroup 6.11-5 AutomorphismGroup
6.11-6 IsomorphicCopyByPerm 6.11-6 QuasigroupIsomorph
6.11-7 IsomorphicCopyByNormalSubloop 6.11-7 LoopIsomorph
6.11-8 Discriminator 6.11-8 IsomorphicCopyByPerm
6.11-9 AreEqualDiscriminators 6.11-9 IsomorphicCopyByNormalSubloop
6.11-10 Discriminator
6.11-11 AreEqualDiscriminators
6.12 Isotopisms 6.12 Isotopisms
6.12-1 IsotopismLoops 6.12-1 IsotopismLoops
6.12-2 LoopsUpToIsotopism 6.12-2 LoopsUpToIsotopism
@ -256,28 +258,31 @@
9.2 Left Bol Loops and Right Bol Loops 9.2 Left Bol Loops and Right Bol Loops
9.2-1 LeftBolLoop 9.2-1 LeftBolLoop
9.2-2 RightBolLoop 9.2-2 RightBolLoop
9.3 Moufang Loops 9.3 Left Bruck Loops and Right Bruck Loops
9.3-1 MoufangLoop 9.3-1 LeftBruckLoop
9.4 Code Loops 9.3-2 RightBruckLoop
9.4-1 CodeLoop 9.4 Moufang Loops
9.5 Steiner Loops 9.4-1 MoufangLoop
9.5-1 SteinerLoop 9.5 Code Loops
9.6 Conjugacy Closed Loops 9.5-1 CodeLoop
9.6-1 RCCLoop and RightConjugacyClosedLoop 9.6 Steiner Loops
9.6-2 LCCLoop and LeftConjugacyClosedLoop 9.6-1 SteinerLoop
9.6-3 CCLoop and ConjugacyClosedLoop 9.7 Conjugacy Closed Loops
9.7 Small Loops 9.7-1 RCCLoop and RightConjugacyClosedLoop
9.7-1 SmallLoop 9.7-2 LCCLoop and LeftConjugacyClosedLoop
9.8 Paige Loops 9.7-3 CCLoop and ConjugacyClosedLoop
9.8-1 PaigeLoop 9.8 Small Loops
9.9 Nilpotent Loops 9.8-1 SmallLoop
9.9-1 NilpotentLoop 9.9 Paige Loops
9.10 Automorphic Loops 9.9-1 PaigeLoop
9.10-1 AutomorphicLoop 9.10 Nilpotent Loops
9.11 Interesting Loops 9.10-1 NilpotentLoop
9.11-1 InterestingLoop 9.11 Automorphic Loops
9.12 Libraries of Loops Up To Isotopism 9.11-1 AutomorphicLoop
9.12-1 ItpSmallLoop 9.12 Interesting Loops
9.12-1 InterestingLoop
9.13 Libraries of Loops Up To Isotopism
9.13-1 ItpSmallLoop
A Files A Files
B Filters B Filters

64
doc/chap0_mj.html
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@ -31,7 +31,7 @@
<h2>Computing with quasigroups and loops in <strong class="pkg">GAP</strong></h2> <h2>Computing with quasigroups and loops in <strong class="pkg">GAP</strong></h2>
<p>Version 3.3.0</p> <p>Version 3.4.0</p>
</div> </div>
<p><b>Gábor P. Nagy <p><b>Gábor P. Nagy
@ -50,7 +50,7 @@
<p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p> <p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p>
<h3>Copyright</h3> <h3>Copyright</h3>
<p>© 2016 Gábor P. Nagy and Petr Vojtěchovský.</p> <p>© 2017 Gábor P. Nagy and Petr Vojtěchovský.</p>
<p><a id="X8537FEB07AF2BEC8" name="X8537FEB07AF2BEC8"></a></p> <p><a id="X8537FEB07AF2BEC8" name="X8537FEB07AF2BEC8"></a></p>
@ -331,10 +331,12 @@
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X82373C5479574F22">6.11-3 QuasigroupsUpToIsomorphism</a></span> <span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X82373C5479574F22">6.11-3 QuasigroupsUpToIsomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X8308F38283C61B20">6.11-4 LoopsUpToIsomorphism</a></span> <span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X8308F38283C61B20">6.11-4 LoopsUpToIsomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X87677B0787B4461A">6.11-5 AutomorphismGroup</a></span> <span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X87677B0787B4461A">6.11-5 AutomorphismGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X85B3E22679FD8D81">6.11-6 IsomorphicCopyByPerm</a></span> <span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X7A42812B7B027DD4">6.11-6 QuasigroupIsomorph</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X8121DE3A78795040">6.11-7 IsomorphicCopyByNormalSubloop</a></span> <span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X7BD1AC32851286EA">6.11-7 LoopIsomorph</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X7D09D8957E4A0973">6.11-8 Discriminator</a></span> <span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X85B3E22679FD8D81">6.11-8 IsomorphicCopyByPerm</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X812F0DEE7C896E18">6.11-9 AreEqualDiscriminators</a></span> <span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X8121DE3A78795040">6.11-9 IsomorphicCopyByNormalSubloop</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X7D09D8957E4A0973">6.11-10 Discriminator</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X812F0DEE7C896E18">6.11-11 AreEqualDiscriminators</a></span>
</div></div> </div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap6_mj.html#X7E996BDD81E594F9">6.12 <span class="Heading">Isotopisms</span></a> <div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap6_mj.html#X7E996BDD81E594F9">6.12 <span class="Heading">Isotopisms</span></a>
</span> </span>
@ -474,60 +476,66 @@
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X7EE99F647C537994">9.2-1 LeftBolLoop</a></span> <span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X7EE99F647C537994">9.2-1 LeftBolLoop</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X8774304282654C58">9.2-2 RightBolLoop</a></span> <span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X8774304282654C58">9.2-2 RightBolLoop</a></span>
</div></div> </div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X7953702D84E60AF4">9.3 <span class="Heading">Moufang Loops</span></a> <div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X8028D69A86B15897">9.3 <span class="Heading">Left Bruck Loops and Right Bruck Loops</span></a>
</span> </span>
<div class="ContSSBlock"> <div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X81E82098822543EE">9.3-1 MoufangLoop</a></span> <span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X8290B01780F0FCD3">9.3-1 LeftBruckLoop</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X798DD7CF871F648F">9.3-2 RightBruckLoop</a></span>
</div></div> </div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X7BCA6BCB847F79DC">9.4 <span class="Heading">Code Loops</span></a> <div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X7953702D84E60AF4">9.4 <span class="Heading">Moufang Loops</span></a>
</span> </span>
<div class="ContSSBlock"> <div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X7DB4D3B27BB4D7EE">9.4-1 CodeLoop</a></span> <span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X81E82098822543EE">9.4-1 MoufangLoop</a></span>
</div></div> </div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X84E941EE7846D3EE">9.5 <span class="Heading">Steiner Loops</span></a> <div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X7BCA6BCB847F79DC">9.5 <span class="Heading">Code Loops</span></a>
</span> </span>
<div class="ContSSBlock"> <div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X87C235457E859AF4">9.5-1 SteinerLoop</a></span> <span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X7DB4D3B27BB4D7EE">9.5-1 CodeLoop</a></span>
</div></div> </div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X867E5F0783FEB8B5">9.6 <span class="Heading">Conjugacy Closed Loops</span></a> <div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X84E941EE7846D3EE">9.6 <span class="Heading">Steiner Loops</span></a>
</span> </span>
<div class="ContSSBlock"> <div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X806B2DE67990E42F">9.6-1 <span class="Heading">RCCLoop and RightConjugacyClosedLoop</span></a> <span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X87C235457E859AF4">9.6-1 SteinerLoop</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X867E5F0783FEB8B5">9.7 <span class="Heading">Conjugacy Closed Loops</span></a>
</span> </span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X80AB8B107D55FB19">9.6-2 <span class="Heading">LCCLoop and LeftConjugacyClosedLoop</span></a> <div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X806B2DE67990E42F">9.7-1 <span class="Heading">RCCLoop and RightConjugacyClosedLoop</span></a>
</span> </span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X798BC601843E8916">9.6-3 <span class="Heading">CCLoop and ConjugacyClosedLoop</span></a> <span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X80AB8B107D55FB19">9.7-2 <span class="Heading">LCCLoop and LeftConjugacyClosedLoop</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X798BC601843E8916">9.7-3 <span class="Heading">CCLoop and ConjugacyClosedLoop</span></a>
</span> </span>
</div></div> </div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X7E3A8F2C790F2CA1">9.7 <span class="Heading">Small Loops</span></a> <div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X7E3A8F2C790F2CA1">9.8 <span class="Heading">Small Loops</span></a>
</span> </span>
<div class="ContSSBlock"> <div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X7C6EE23E84CD87D3">9.7-1 SmallLoop</a></span> <span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X7C6EE23E84CD87D3">9.8-1 SmallLoop</a></span>
</div></div> </div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X8135C8FD8714C606">9.8 <span class="Heading">Paige Loops</span></a> <div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X8135C8FD8714C606">9.9 <span class="Heading">Paige Loops</span></a>
</span> </span>
<div class="ContSSBlock"> <div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X7FCF4D6B7AD66D74">9.8-1 PaigeLoop</a></span> <span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X7FCF4D6B7AD66D74">9.9-1 PaigeLoop</a></span>
</div></div> </div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X86695C577A4D1784">9.9 <span class="Heading">Nilpotent Loops</span></a> <div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X86695C577A4D1784">9.10 <span class="Heading">Nilpotent Loops</span></a>
</span> </span>
<div class="ContSSBlock"> <div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X7A9C960D86E2AD28">9.9-1 NilpotentLoop</a></span> <span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X7A9C960D86E2AD28">9.10-1 NilpotentLoop</a></span>
</div></div> </div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X793B22EA8643C667">9.10 <span class="Heading">Automorphic Loops</span></a> <div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X793B22EA8643C667">9.11 <span class="Heading">Automorphic Loops</span></a>
</span> </span>
<div class="ContSSBlock"> <div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X784FFA9E7FDA9F43">9.10-1 AutomorphicLoop</a></span> <span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X784FFA9E7FDA9F43">9.11-1 AutomorphicLoop</a></span>
</div></div> </div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X843BD73F788049F7">9.11 <span class="Heading">Interesting Loops</span></a> <div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X843BD73F788049F7">9.12 <span class="Heading">Interesting Loops</span></a>
</span> </span>
<div class="ContSSBlock"> <div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X87F24AD3811910D3">9.11-1 InterestingLoop</a></span> <span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X87F24AD3811910D3">9.12-1 InterestingLoop</a></span>
</div></div> </div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X864839227D5C0A90">9.12 <span class="Heading">Libraries of Loops Up To Isotopism</span></a> <div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X864839227D5C0A90">9.13 <span class="Heading">Libraries of Loops Up To Isotopism</span></a>
</span> </span>
<div class="ContSSBlock"> <div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X850C4C01817A098D">9.12-1 ItpSmallLoop</a></span> <span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X850C4C01817A098D">9.13-1 ItpSmallLoop</a></span>
</div></div> </div></div>
</div> </div>
<div class="ContChap"><a href="chapA_mj.html#X7BC4571A79FFB7D0">A <span class="Heading">Files</span></a> <div class="ContChap"><a href="chapA_mj.html#X7BC4571A79FFB7D0">A <span class="Heading">Files</span></a>

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@ -20,7 +20,7 @@
1.2 Installation 1.2 Installation
Have GAP 4.7 or newer installed on your computer. Have GAP 4.8 or newer installed on your computer.
If you do not see the subfolder pkg/loops in the main directory of GAP then If you do not see the subfolder pkg/loops in the main directory of GAP then
download the LOOPS package from the distribution website download the LOOPS package from the distribution website
@ -85,14 +85,15 @@
We thank the following people for sending us remarks and comments, and for We thank the following people for sending us remarks and comments, and for
suggesting new functionality of the package: Muniru Asiru, Bjoern Assmann, suggesting new functionality of the package: Muniru Asiru, Bjoern Assmann,
Andreas Distler, Aleš Drápal, Steve Flammia, Kenneth W. Johnson, Michael K. Andreas Distler, Aleš Drápal, Graham Ellis, Steve Flammia, Kenneth W.
Kinyon, Alexander Konovalov, Frank Lübeck and Jonathan D.H. Smith. Johnson, Michael K. Kinyon, Alexander Konovalov, Frank Lübeck, Jonathan D.H.
Smith, David Stanovský and Glen Whitney.
The library of Moufang loops of order 243 was generated from data provided The library of Moufang loops of order 243 was generated from data provided
by Michael C. Slattery and Ashley L. Zenisek. The library of right conjugacy by Michael C. Slattery and Ashley L. Zenisek. The library of right conjugacy
closed loops of order less than 28 was generated from data provided by closed loops of order less than 28 was generated from data provided by
Katharina Artic. The library of commutative automorphic loops of order 27, Katharina Artic. The library of right Bruck loops of order 27, 81 was
81 and 243 was obtained jointly with Izabella Stuhl. obtained jointly with Izabella Stuhl.
Gábor P. Nagy was supported by OTKA grants F042959 and T043758, and Petr Gábor P. Nagy was supported by OTKA grants F042959 and T043758, and Petr
Vojtěchovský was supported by the 2006 and 2016 University of Denver PROF Vojtěchovský was supported by the 2006 and 2016 University of Denver PROF

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@ -73,7 +73,7 @@
<h4>1.2 <span class="Heading">Installation</span></h4> <h4>1.2 <span class="Heading">Installation</span></h4>
<p>Have <strong class="pkg">GAP 4.7</strong> or newer installed on your computer.</p> <p>Have <strong class="pkg">GAP 4.8</strong> or newer installed on your computer.</p>
<p>If you do not see the subfolder <code class="file">pkg/loops</code> in the main directory of <strong class="pkg">GAP</strong> then download the <strong class="pkg">LOOPS</strong> package from the distribution website <span class="URL"><a href="http://www.math.du.edu/loops">http://www.math.du.edu/loops</a></span> and unpack the downloaded file into the <code class="file">pkg</code> subfolder.</p> <p>If you do not see the subfolder <code class="file">pkg/loops</code> in the main directory of <strong class="pkg">GAP</strong> then download the <strong class="pkg">LOOPS</strong> package from the distribution website <span class="URL"><a href="http://www.math.du.edu/loops">http://www.math.du.edu/loops</a></span> and unpack the downloaded file into the <code class="file">pkg</code> subfolder.</p>
@ -127,9 +127,9 @@ gap&gt; WriteGapIniFile();;
<h4>1.7 <span class="Heading">Acknowledgment</span></h4> <h4>1.7 <span class="Heading">Acknowledgment</span></h4>
<p>We thank the following people for sending us remarks and comments, and for suggesting new functionality of the package: Muniru Asiru, Bjoern Assmann, Andreas Distler, Aleš Drápal, Steve Flammia, Kenneth W. Johnson, Michael K. Kinyon, Alexander Konovalov, Frank Lübeck and Jonathan D.H. Smith.</p> <p>We thank the following people for sending us remarks and comments, and for suggesting new functionality of the package: Muniru Asiru, Bjoern Assmann, Andreas Distler, Aleš Drápal, Graham Ellis, Steve Flammia, Kenneth W. Johnson, Michael K. Kinyon, Alexander Konovalov, Frank Lübeck, Jonathan D.H. Smith, David Stanovský and Glen Whitney.</p>
<p>The library of Moufang loops of order 243 was generated from data provided by Michael C. Slattery and Ashley L. Zenisek. The library of right conjugacy closed loops of order less than 28 was generated from data provided by Katharina Artic. The library of commutative automorphic loops of order 27, 81 and 243 was obtained jointly with Izabella Stuhl.</p> <p>The library of Moufang loops of order 243 was generated from data provided by Michael C. Slattery and Ashley L. Zenisek. The library of right conjugacy closed loops of order less than 28 was generated from data provided by Katharina Artic. The library of right Bruck loops of order 27, 81 was obtained jointly with Izabella Stuhl.</p>
<p>Gábor P. Nagy was supported by OTKA grants F042959 and T043758, and Petr Vojtěchovský was supported by the 2006 and 2016 University of Denver PROF grants and the Simons Foundation Collaboration Grant 210176.</p> <p>Gábor P. Nagy was supported by OTKA grants F042959 and T043758, and Petr Vojtěchovský was supported by the 2006 and 2016 University of Denver PROF grants and the Simons Foundation Collaboration Grant 210176.</p>

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@ -40,8 +40,8 @@
DeclareRepresentation( "IsLoopElmRep", DeclareRepresentation( "IsLoopElmRep",
IsPositionalObjectRep and IsMultiplicativeElementWithInverse, [1] ); IsPositionalObjectRep and IsMultiplicativeElementWithInverse, [1] );
## latin (auxiliary category for GAP to tell apart IsMagma and IsQuasigroup) ## latin (auxiliary category for GAP to tell apart IsMagma and IsQuasigroup)
DeclareCategory( "IsLatin", IsObject ); DeclareCategory( "IsLatinMagma", IsObject );
DeclareCategory( "IsQuasigroup", IsMagma and IsLatin ); DeclareCategory( "IsQuasigroup", IsMagma and IsLatinMagma );
DeclareCategory( "IsLoop", IsQuasigroup and DeclareCategory( "IsLoop", IsQuasigroup and
IsMultiplicativeElementWithInverseCollection); IsMultiplicativeElementWithInverseCollection);

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@ -81,8 +81,8 @@ DeclareCategory( "IsLoopElement",
DeclareRepresentation( "IsLoopElmRep", DeclareRepresentation( "IsLoopElmRep",
IsPositionalObjectRep and IsMultiplicativeElementWithInverse, [1] ); IsPositionalObjectRep and IsMultiplicativeElementWithInverse, [1] );
## latin (auxiliary category for GAP to tell apart IsMagma and IsQuasigroup) ## latin (auxiliary category for GAP to tell apart IsMagma and IsQuasigroup)
DeclareCategory( "IsLatin", IsObject ); DeclareCategory( "IsLatinMagma", IsObject );
DeclareCategory( "IsQuasigroup", IsMagma and IsLatin ); DeclareCategory( "IsQuasigroup", IsMagma and IsLatinMagma );
DeclareCategory( "IsLoop", IsQuasigroup and DeclareCategory( "IsLoop", IsQuasigroup and
IsMultiplicativeElementWithInverseCollection); IsMultiplicativeElementWithInverseCollection);
@ -163,13 +163,13 @@ DeclareCategory( "IsLoop", IsQuasigroup and
<div class="example"><pre> <div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">L; </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">L;</span>
&lt;loop of order 2&gt; &lt;loop of order 2&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Print( L ); </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Print( L );</span>
&lt;loop with multiplication table [ [ 1, 2 ], [ 2, 1 ] ]&gt; &lt;loop with multiplication table [ [ 1, 2 ], [ 2, 1 ] ]&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Elements( L ); </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Elements( L );</span>
[ l1, l2 ] [ l1, l2 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">SetLoopElmName( L, "loop_element" );; Elements( L ); </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">SetLoopElmName( L, "loop_element" );; Elements( L );</span>
[ loop_element1, loop_element2 ] [ loop_element1, loop_element2 ]
</pre></div> </pre></div>

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@ -189,13 +189,13 @@
<div class="example"><pre> <div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">ct := CanonicalCayleyTable( [[5,3],[3,5]] ); </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">ct := CanonicalCayleyTable( [[5,3],[3,5]] );</span>
[ [ 2, 1 ], [ 1, 2 ] ] [ [ 2, 1 ], [ 1, 2 ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">NormalizedQuasigroupTable( ct ); </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">NormalizedQuasigroupTable( ct );</span>
[ [ 1, 2 ], [ 2, 1 ] ] [ [ 1, 2 ], [ 2, 1 ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LoopByCayleyTable( last ); </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LoopByCayleyTable( last );</span>
&lt;loop of order 2&gt; &lt;loop of order 2&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ IsQuasigroupTable( ct ), IsLoopTable( ct ) ]; </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ IsQuasigroupTable( ct ), IsLoopTable( ct ) ];</span>
[ true, false ] [ true, false ]
</pre></div> </pre></div>
@ -330,12 +330,12 @@
<div class="example"><pre> <div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">S := Subloop( MoufangLoop( 12, 1 ), [ 3 ] );; </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">S := Subloop( MoufangLoop( 12, 1 ), [ 3 ] );;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">ls := LeftSection( S ); </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">ls := LeftSection( S );</span>
[ (), (1,3,5), (1,5,3) ] [ (), (1,3,5), (1,5,3) ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">CayleyTableByPerms( ls ); </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">CayleyTableByPerms( ls );</span>
[ [ 1, 3, 5 ], [ 3, 5, 1 ], [ 5, 1, 3 ] ] [ [ 1, 3, 5 ], [ 3, 5, 1 ], [ 5, 1, 3 ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">CayleyTable( LoopByLeftSection( ls ) ); </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">CayleyTable( LoopByLeftSection( ls ) );</span>
[ [ 1, 2, 3 ], [ 2, 3, 1 ], [ 3, 1, 2 ] ] [ [ 1, 2, 3 ], [ 2, 3, 1 ], [ 3, 1, 2 ] ]
</pre></div> </pre></div>
@ -361,9 +361,9 @@
<div class="example"><pre> <div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">T := [ (), (1,2)(3,4,5), (1,3,5)(2,4), (1,4,3)(2,5), (1,5,4)(2,3) ];; </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">T := [ (), (1,2)(3,4,5), (1,3,5)(2,4), (1,4,3)(2,5), (1,5,4)(2,3) ];;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">G := Group( T );; H := Stabilizer( G, 1 );; </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">G := Group( T );; H := Stabilizer( G, 1 );;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LoopByRightFolder( G, H, T ); </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LoopByRightFolder( G, H, T );</span>
&lt;loop of order 5&gt; &lt;loop of order 5&gt;
</pre></div> </pre></div>
@ -395,14 +395,14 @@
<div class="example"><pre> <div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">F := IntoLoop( Group( (1,2) ) ); </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">F := IntoLoop( Group( (1,2) ) );</span>
&lt;loop of order 2&gt; &lt;loop of order 2&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">K := DirectProduct( F, F );; </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">K := DirectProduct( F, F );;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">phi := [ (), (2,3) ];; </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">phi := [ (), (2,3) ];;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">theta := [ [ 1, 1 ], [ 1, 3 ] ];; </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">theta := [ [ 1, 1 ], [ 1, 3 ] ];;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LoopByExtension( K, F, phi, theta ); </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LoopByExtension( K, F, phi, theta );</span>
&lt;loop of order 8&gt; &lt;loop of order 8&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">IsAssociative( last ); </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">IsAssociative( last );</span>
false false
</pre></div> </pre></div>

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@ -168,15 +168,15 @@
<div class="example"><pre> <div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">CayleyTable( Q ); </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">CayleyTable( Q );</span>
[ [ 1, 2, 3, 4, 5 ], [ [ 1, 2, 3, 4, 5 ],
[ 2, 1, 4, 5, 3 ], [ 2, 1, 4, 5, 3 ],
[ 3, 4, 5, 1, 2 ], [ 3, 4, 5, 1, 2 ],
[ 4, 5, 2, 3, 1 ], [ 4, 5, 2, 3, 1 ],
[ 5, 3, 1, 2, 4 ] ] [ 5, 3, 1, 2, 4 ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">elms := Elements( Q ); </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">elms := Elements( Q );</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ l1, l2, l3, l4, l5 ]; </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ l1, l2, l3, l4, l5 ];</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ LeftInverse( elms[3] ), RightInverse( elms[3] ), Inverse( elms[3] ) ]; </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ LeftInverse( elms[3] ), RightInverse( elms[3] ), Inverse( elms[3] ) ];</span>
[ l5, l4, fail ] [ l5, l4, fail ]
</pre></div> </pre></div>

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@ -488,17 +488,28 @@
the elements of the underlying quasigroup without changing the isomorphism the elements of the underlying quasigroup without changing the isomorphism
type of the quasigroups. LOOPS contains several functions for this purpose. type of the quasigroups. LOOPS contains several functions for this purpose.
6.11-6 IsomorphicCopyByPerm 6.11-6 QuasigroupIsomorph
QuasigroupIsomorph( Q, f )  operation
Returns: When Q is a quasigroup and f is a permutation of 1,dots,|Q|,
returns the quasigroup defined on the same set as Q with
multiplication * defined by x*y =f(f^-1(x)f^-1(y)).
6.11-7 LoopIsomorph
LoopIsomorph( Q, f )  operation
Returns: When Q is a loop and f is a permutation of 1,dots,|Q| fixing 1,
returns the loop defined on the same set as Q with multiplication
* defined by x*y =f(f^-1(x)f^-1(y)). If f(1)=cne 1, the
isomorphism (1,c) is applied after f.
6.11-8 IsomorphicCopyByPerm
IsomorphicCopyByPerm( Q, f )  operation IsomorphicCopyByPerm( Q, f )  operation
Returns: When Q is a quasigroup and f is a permutation of 1,dots,|Q|, Returns: LoopIsomorphism(Q,f) if Q is a loop, and
returns a quasigroup defined on the same set as Q with QuasigroupIsomorphism(Q,f) if Q is a quasigroup.
multiplication * defined by x*y =f(f^-1(x)f^-1(y)). When Q is a
declared loop, a loop is returned. Consequently, when Q is a
declared loop and f(1) = kne 1, then f is first replaced with f∘
(1,k), to make sure that the resulting Cayley table is normalized.
6.11-7 IsomorphicCopyByNormalSubloop 6.11-9 IsomorphicCopyByNormalSubloop
IsomorphicCopyByNormalSubloop( Q, S )  operation IsomorphicCopyByNormalSubloop( Q, S )  operation
Returns: When S is a normal subloop of a loop Q, returns an isomorphic copy Returns: When S is a normal subloop of a loop Q, returns an isomorphic copy
@ -511,7 +522,7 @@
these invariants to partition the loop into blocks of elements preserved these invariants to partition the loop into blocks of elements preserved
under isomorphisms. The following two operations are used in the search. under isomorphisms. The following two operations are used in the search.
6.11-8 Discriminator 6.11-10 Discriminator
Discriminator( Q )  operation Discriminator( Q )  operation
Returns: A data structure with isomorphism invariants of a loop Q. Returns: A data structure with isomorphism invariants of a loop Q.
@ -523,7 +534,7 @@
If two loops have different discriminators, they are not isomorphic. If they If two loops have different discriminators, they are not isomorphic. If they
have identical discriminators, they may or may not be isomorphic. have identical discriminators, they may or may not be isomorphic.
6.11-9 AreEqualDiscriminators 6.11-11 AreEqualDiscriminators
AreEqualDiscriminators( D1, D2 )  operation AreEqualDiscriminators( D1, D2 )  operation
Returns: true if D1, D2 are equal discriminators for the purposes of Returns: true if D1, D2 are equal discriminators for the purposes of

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@ -120,10 +120,12 @@
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X82373C5479574F22">6.11-3 QuasigroupsUpToIsomorphism</a></span> <span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X82373C5479574F22">6.11-3 QuasigroupsUpToIsomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X8308F38283C61B20">6.11-4 LoopsUpToIsomorphism</a></span> <span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X8308F38283C61B20">6.11-4 LoopsUpToIsomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X87677B0787B4461A">6.11-5 AutomorphismGroup</a></span> <span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X87677B0787B4461A">6.11-5 AutomorphismGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X85B3E22679FD8D81">6.11-6 IsomorphicCopyByPerm</a></span> <span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X7A42812B7B027DD4">6.11-6 QuasigroupIsomorph</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X8121DE3A78795040">6.11-7 IsomorphicCopyByNormalSubloop</a></span> <span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X7BD1AC32851286EA">6.11-7 LoopIsomorph</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X7D09D8957E4A0973">6.11-8 Discriminator</a></span> <span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X85B3E22679FD8D81">6.11-8 IsomorphicCopyByPerm</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X812F0DEE7C896E18">6.11-9 AreEqualDiscriminators</a></span> <span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X8121DE3A78795040">6.11-9 IsomorphicCopyByNormalSubloop</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X7D09D8957E4A0973">6.11-10 Discriminator</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X812F0DEE7C896E18">6.11-11 AreEqualDiscriminators</a></span>
</div></div> </div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap6_mj.html#X7E996BDD81E594F9">6.12 <span class="Heading">Isotopisms</span></a> <div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap6_mj.html#X7E996BDD81E594F9">6.12 <span class="Heading">Isotopisms</span></a>
</span> </span>
@ -266,26 +268,26 @@
<div class="example"><pre> <div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">M := MoufangLoop( 12, 1 );; S := Subloop( M, [ M.5 ] ); </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">M := MoufangLoop( 12, 1 );; S := Subloop( M, [ M.5 ] );</span>
&lt;loop of order 3&gt; &lt;loop of order 3&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ Parent( S ) = M, Elements( S ), PosInParent( S ) ]; </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ Parent( S ) = M, Elements( S ), PosInParent( S ) ];</span>
[ true, [ l1, l3, l5], [ 1, 3, 5 ] ] [ true, [ l1, l3, l5], [ 1, 3, 5 ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">HasCayleyTable( S ); </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">HasCayleyTable( S );</span>
false false
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">SetLoopElmName( S, "s" );; Elements( S ); Elements( M ); </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">SetLoopElmName( S, "s" );; Elements( S ); Elements( M );</span>
[ s1, s3, s5 ] [ s1, s3, s5 ]
[ s1, s2, s3, s4, s5, s6, s7, s8, s9, s10, s11, s12 ] [ s1, s2, s3, s4, s5, s6, s7, s8, s9, s10, s11, s12 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">CayleyTable( S ); </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">CayleyTable( S );</span>
[ [ 1, 3, 5 ], [ 3, 5, 1 ], [ 5, 1, 3 ] ] [ [ 1, 3, 5 ], [ 3, 5, 1 ], [ 5, 1, 3 ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LeftSection( S ); </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LeftSection( S );</span>
[ (), (1,3,5), (1,5,3) ] [ (), (1,3,5), (1,5,3) ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ HasCayleyTable( S ), Parent( S ) = M ]; </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ HasCayleyTable( S ), Parent( S ) = M ];</span>
[ true, true ] [ true, true ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">L := LoopByCayleyTable( CayleyTable( S ) );; Elements( L ); </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">L := LoopByCayleyTable( CayleyTable( S ) );; Elements( L );</span>
[ l1, l2, l3 ] [ l1, l2, l3 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ Parent( L ) = L, IsSubloop( M, S ), IsSubloop( M, L ) ]; </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ Parent( L ) = L, IsSubloop( M, S ), IsSubloop( M, L ) ];</span>
[ true, true, false ] [ true, true, false ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LeftSection( L ); </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LeftSection( L );</span>
[ (), (1,2,3), (1,3,2) ] [ (), (1,2,3), (1,3,2) ]
</pre></div> </pre></div>
@ -350,14 +352,14 @@ false
<div class="example"><pre> <div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">M := MoufangLoop(12,1); </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">M := MoufangLoop(12,1);</span>
&lt;Moufang loop 12/1&gt; &lt;Moufang loop 12/1&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LeftSection(M)[2]; </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LeftSection(M)[2];</span>
(1,2)(3,4)(5,6)(7,8)(9,12)(10,11) (1,2)(3,4)(5,6)(7,8)(9,12)(10,11)
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Mlt := MultiplicationGroup(M); Inn := InnerMappingGroup(M); </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Mlt := MultiplicationGroup(M); Inn := InnerMappingGroup(M);</span>
&lt;permutation group of size 2592 with 23 generators&gt; &lt;permutation group of size 2592 with 23 generators&gt;
Group([ (4,6)(7,11), (7,11)(8,10), (2,6,4)(7,9,11), (3,5)(9,11), (8,12,10) ]) Group([ (4,6)(7,11), (7,11)(8,10), (2,6,4)(7,9,11), (3,5)(9,11), (8,12,10) ])
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Size(Inn); </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Size(Inn);</span>
216 216
</pre></div> </pre></div>
@ -465,13 +467,13 @@ Group([ (4,6)(7,11), (7,11)(8,10), (2,6,4)(7,9,11), (3,5)(9,11), (8,12,10) ])
<div class="example"><pre> <div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">M := MoufangLoop( 12, 1 );; S := Subloop( M, [ M.3 ] ); </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">M := MoufangLoop( 12, 1 );; S := Subloop( M, [ M.3 ] );</span>
&lt;loop of order 3&gt; &lt;loop of order 3&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">IsNormal( M, S ); </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">IsNormal( M, S );</span>
true true
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">F := FactorLoop( M, S ); </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">F := FactorLoop( M, S );</span>
&lt;loop of order 4&gt; &lt;loop of order 4&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">NaturalHomomorphismByNormalSubloop( M, S ); </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">NaturalHomomorphismByNormalSubloop( M, S );</span>
MappingByFunction( &lt;loop of order 12&gt;, &lt;loop of order 4&gt;, MappingByFunction( &lt;loop of order 12&gt;, &lt;loop of order 4&gt;,
function( x ) ... end ) function( x ) ... end )
</pre></div> </pre></div>
@ -607,16 +609,30 @@ MappingByFunction( &lt;loop of order 12&gt;, &lt;loop of order 4&gt;,
<p>While dealing with Cayley tables, it is often useful to rename or reorder the elements of the underlying quasigroup without changing the isomorphism type of the quasigroups. <strong class="pkg">LOOPS</strong> contains several functions for this purpose.</p> <p>While dealing with Cayley tables, it is often useful to rename or reorder the elements of the underlying quasigroup without changing the isomorphism type of the quasigroups. <strong class="pkg">LOOPS</strong> contains several functions for this purpose.</p>
<p><a id="X7A42812B7B027DD4" name="X7A42812B7B027DD4"></a></p>
<h5>6.11-6 QuasigroupIsomorph</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; QuasigroupIsomorph</code>( <var class="Arg">Q</var>, <var class="Arg">f</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: When <var class="Arg">Q</var> is a quasigroup and <var class="Arg">f</var> is a permutation of <span class="SimpleMath">\(1,\dots,|\)</span><var class="Arg">Q</var><span class="SimpleMath">\(|\)</span>, returns the quasigroup defined on the same set as <var class="Arg">Q</var> with multiplication <span class="SimpleMath">\(*\)</span> defined by <span class="SimpleMath">\(x*y = \)</span><var class="Arg">f</var><span class="SimpleMath">\((\)</span><var class="Arg">f</var><span class="SimpleMath">\({}^{-1}(x)\)</span><var class="Arg">f</var><span class="SimpleMath">\({}^{-1}(y))\)</span>.</p>
<p><a id="X7BD1AC32851286EA" name="X7BD1AC32851286EA"></a></p>
<h5>6.11-7 LoopIsomorph</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; LoopIsomorph</code>( <var class="Arg">Q</var>, <var class="Arg">f</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: When <var class="Arg">Q</var> is a loop and <var class="Arg">f</var> is a permutation of <span class="SimpleMath">\(1,\dots,|\)</span><var class="Arg">Q</var><span class="SimpleMath">\(|\)</span> fixing <span class="SimpleMath">\(1\)</span>, returns the loop defined on the same set as <var class="Arg">Q</var> with multiplication <span class="SimpleMath">\(*\)</span> defined by <span class="SimpleMath">\(x*y = \)</span><var class="Arg">f</var><span class="SimpleMath">\((\)</span><var class="Arg">f</var><span class="SimpleMath">\({}^{-1}(x)\)</span><var class="Arg">f</var><span class="SimpleMath">\({}^{-1}(y))\)</span>. If <var class="Arg">f</var><span class="SimpleMath">\((1)=c\ne 1\)</span>, the isomorphism <span class="SimpleMath">\((1,c)\)</span> is applied after <var class="Arg">f</var>.</p>
<p><a id="X85B3E22679FD8D81" name="X85B3E22679FD8D81"></a></p> <p><a id="X85B3E22679FD8D81" name="X85B3E22679FD8D81"></a></p>
<h5>6.11-6 IsomorphicCopyByPerm</h5> <h5>6.11-8 IsomorphicCopyByPerm</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsomorphicCopyByPerm</code>( <var class="Arg">Q</var>, <var class="Arg">f</var> )</td><td class="tdright">( operation )</td></tr></table></div> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsomorphicCopyByPerm</code>( <var class="Arg">Q</var>, <var class="Arg">f</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: When <var class="Arg">Q</var> is a quasigroup and <var class="Arg">f</var> is a permutation of <span class="SimpleMath">\(1,\dots,|\)</span><var class="Arg">Q</var><span class="SimpleMath">\(|\)</span>, returns a quasigroup defined on the same set as <var class="Arg">Q</var> with multiplication <span class="SimpleMath">\(*\)</span> defined by <span class="SimpleMath">\(x*y = \)</span><var class="Arg">f</var><span class="SimpleMath">\((\)</span><var class="Arg">f</var><span class="SimpleMath">\({}^{-1}(x)\)</span><var class="Arg">f</var><span class="SimpleMath">\({}^{-1}(y))\)</span>. When <var class="Arg">Q</var> is a declared loop, a loop is returned. Consequently, when <var class="Arg">Q</var> is a declared loop and <var class="Arg">f</var><span class="SimpleMath">\((1) = k\ne 1\)</span>, then <var class="Arg">f</var> is first replaced with <var class="Arg">f</var><span class="SimpleMath">\(\circ (1,k)\)</span>, to make sure that the resulting Cayley table is normalized.</p> <p>Returns: <code class="code">LoopIsomorphism(<var class="Arg">Q</var>,<var class="Arg">f</var>)</code> if <var class="Arg">Q</var> is a loop, and <code class="code">QuasigroupIsomorphism(<var class="Arg">Q</var>,<var class="Arg">f</var>)</code> if <var class="Arg">Q</var> is a quasigroup.</p>
<p><a id="X8121DE3A78795040" name="X8121DE3A78795040"></a></p> <p><a id="X8121DE3A78795040" name="X8121DE3A78795040"></a></p>
<h5>6.11-7 IsomorphicCopyByNormalSubloop</h5> <h5>6.11-9 IsomorphicCopyByNormalSubloop</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsomorphicCopyByNormalSubloop</code>( <var class="Arg">Q</var>, <var class="Arg">S</var> )</td><td class="tdright">( operation )</td></tr></table></div> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsomorphicCopyByNormalSubloop</code>( <var class="Arg">Q</var>, <var class="Arg">S</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: When <var class="Arg">S</var> is a normal subloop of a loop <var class="Arg">Q</var>, returns an isomorphic copy of <var class="Arg">Q</var> in which the elements are ordered according to the right cosets of <var class="Arg">S</var>. In particular, the Cayley table of <var class="Arg">S</var> will appear in the top left corner of the Cayley table of the resulting loop.<br /></p> <p>Returns: When <var class="Arg">S</var> is a normal subloop of a loop <var class="Arg">Q</var>, returns an isomorphic copy of <var class="Arg">Q</var> in which the elements are ordered according to the right cosets of <var class="Arg">S</var>. In particular, the Cayley table of <var class="Arg">S</var> will appear in the top left corner of the Cayley table of the resulting loop.<br /></p>
@ -625,7 +641,7 @@ MappingByFunction( &lt;loop of order 12&gt;, &lt;loop of order 4&gt;,
<p><a id="X7D09D8957E4A0973" name="X7D09D8957E4A0973"></a></p> <p><a id="X7D09D8957E4A0973" name="X7D09D8957E4A0973"></a></p>
<h5>6.11-8 Discriminator</h5> <h5>6.11-10 Discriminator</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Discriminator</code>( <var class="Arg">Q</var> )</td><td class="tdright">( operation )</td></tr></table></div> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Discriminator</code>( <var class="Arg">Q</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: A data structure with isomorphism invariants of a loop <var class="Arg">Q</var>.</p> <p>Returns: A data structure with isomorphism invariants of a loop <var class="Arg">Q</var>.</p>
@ -636,7 +652,7 @@ MappingByFunction( &lt;loop of order 12&gt;, &lt;loop of order 4&gt;,
<p><a id="X812F0DEE7C896E18" name="X812F0DEE7C896E18"></a></p> <p><a id="X812F0DEE7C896E18" name="X812F0DEE7C896E18"></a></p>
<h5>6.11-9 AreEqualDiscriminators</h5> <h5>6.11-11 AreEqualDiscriminators</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; AreEqualDiscriminators</code>( <var class="Arg">D1</var>, <var class="Arg">D2</var> )</td><td class="tdright">( operation )</td></tr></table></div> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; AreEqualDiscriminators</code>( <var class="Arg">D1</var>, <var class="Arg">D2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: <code class="code">true</code> if <var class="Arg">D1</var>, <var class="Arg">D2</var> are equal discriminators for the purposes of isomorphism searches.</p> <p>Returns: <code class="code">true</code> if <var class="Arg">D1</var>, <var class="Arg">D2</var> are equal discriminators for the purposes of isomorphism searches.</p>

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@ -445,17 +445,17 @@
<div class="example"><pre> <div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">L := LoopByCayleyTable( [ [ 1, 2 ], [ 2, 1 ] ] ); </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">L := LoopByCayleyTable( [ [ 1, 2 ], [ 2, 1 ] ] );</span>
&lt;loop of order 2&gt; &lt;loop of order 2&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ IsLeftBolLoop( L ), L ] </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ IsLeftBolLoop( L ), L ]</span>
[ true, &lt;left Bol loop of order 2&gt; ] [ true, &lt;left Bol loop of order 2&gt; ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ HasIsLeftAlternativeLoop( L ), IsLeftAlternativeLoop( L ) ]; </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ HasIsLeftAlternativeLoop( L ), IsLeftAlternativeLoop( L ) ];</span>
[ true, true ] [ true, true ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ HasIsRightBolLoop( L ), IsRightBolLoop( L ) ]; </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ HasIsRightBolLoop( L ), IsRightBolLoop( L ) ];</span>
[ false, true ] [ false, true ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">L; </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">L;</span>
&lt;Moufang loop of order 2&gt; &lt;Moufang loop of order 2&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ IsAssociative( L ), L ]; </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ IsAssociative( L ), L ];</span>
[ true, &lt;associative loop of order 2&gt; ] [ true, &lt;associative loop of order 2&gt; ]
</pre></div> </pre></div>

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@ -203,11 +203,11 @@
<div class="example"><pre> <div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=PGL(3,3); </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=PGL(3,3);</span>
Group([ (6,7)(8,11)(9,13)(10,12), (1,2,5,7,13,3,8,6,10,9,12,4,11) ]) Group([ (6,7)(8,11)(9,13)(10,12), (1,2,5,7,13,3,8,6,10,9,12,4,11) ])
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">a:=AllLoopTablesInGroup(g,3,0);; Size(a); </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">a:=AllLoopTablesInGroup(g,3,0);; Size(a);</span>
56 56
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">a:=AllLoopsWithMltGroup(g,3,0);; Size(a); </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">a:=AllLoopsWithMltGroup(g,3,0);; Size(a);</span>
52 52
</pre></div> </pre></div>

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@ -78,12 +78,34 @@
retrieved by calling Opposite on left Bol loops. retrieved by calling Opposite on left Bol loops.
9.3 Moufang Loops 9.3 Left Bruck Loops and Right Bruck Loops
The emmerging library named left Bruck contains all left Bruck loops of
orders 3, 9, 27 and 81 (there are 1, 2, 7 and 72 such loops, respectively).
For an odd prime p, left Bruck loops of order p^k are centrally nilpotent
and hence central extensions of the cyclic group of order p by a left Bruck
loop of order p^k-1. It is known that left Bruck loops of order p and p^2
are abelian groups; we have included them in the library because of the
iterative nature of the construction of nilpotent loops.
9.3-1 LeftBruckLoop
LeftBruckLoop( n, m )  function
Returns: The mth left Bruck loop of order n in the library.
9.3-2 RightBruckLoop
RightBruckLoop( n, m )  function
Returns: The mth right Bruck loop of order n in the library.
9.4 Moufang Loops
The library named Moufang contains all nonassociative Moufang loops of order The library named Moufang contains all nonassociative Moufang loops of order
nle 64 and n∈{81,243}. nle 64 and n∈{81,243}.
9.3-1 MoufangLoop 9.4-1 MoufangLoop
MoufangLoop( n, m )  function MoufangLoop( n, m )  function
Returns: The mth Moufang loop of order n in the library. Returns: The mth Moufang loop of order n in the library.
@ -107,20 +129,20 @@
obtained as MoufangLoop(16,3). obtained as MoufangLoop(16,3).
9.4 Code Loops 9.5 Code Loops
The library named code contains all nonassociative code loops of order less The library named code contains all nonassociative code loops of order less
than 65. There are 5 such loops of order 16, 16 of order 32, and 80 of order than 65. There are 5 such loops of order 16, 16 of order 32, and 80 of order
64, all Moufang. The library merely points to the corresponding Moufang 64, all Moufang. The library merely points to the corresponding Moufang
loops. See [NV07] for a classification of small code loops. loops. See [NV07] for a classification of small code loops.
9.4-1 CodeLoop 9.5-1 CodeLoop
CodeLoop( n, m )  function CodeLoop( n, m )  function
Returns: The mth code loop of order n in the library. Returns: The mth code loop of order n in the library.
9.5 Steiner Loops 9.6 Steiner Loops
Here is how the libary named Steiner is described within LOOPS: Here is how the libary named Steiner is described within LOOPS:
@ -141,13 +163,13 @@
Our labeling of Steiner loops of order 16 coincides with the labeling of Our labeling of Steiner loops of order 16 coincides with the labeling of
Steiner triple systems of order 15 in [CR99]. Steiner triple systems of order 15 in [CR99].
9.5-1 SteinerLoop 9.6-1 SteinerLoop
SteinerLoop( n, m )  function SteinerLoop( n, m )  function
Returns: The mth Steiner loop of order n in the library. Returns: The mth Steiner loop of order n in the library.
9.6 Conjugacy Closed Loops 9.7 Conjugacy Closed Loops
The library named RCC contains all nonassocitive right conjugacy closed The library named RCC contains all nonassocitive right conjugacy closed
loops of order nle 27 up to isomorphism. The data for the library was loops of order nle 27 up to isomorphism. The data for the library was
@ -171,14 +193,14 @@
9.6-1 RCCLoop and RightConjugacyClosedLoop 9.7-1 RCCLoop and RightConjugacyClosedLoop
RCCLoop( n, m )  function RCCLoop( n, m )  function
RightConjugacyClosedLoop( n, m )  function RightConjugacyClosedLoop( n, m )  function
Returns: The mth right conjugacy closed loop of order n in the library. Returns: The mth right conjugacy closed loop of order n in the library.
9.6-2 LCCLoop and LeftConjugacyClosedLoop 9.7-2 LCCLoop and LeftConjugacyClosedLoop
LCCLoop( n, m )  function LCCLoop( n, m )  function
LeftConjugacyClosedLoop( n, m )  function LeftConjugacyClosedLoop( n, m )  function
@ -188,8 +210,10 @@
Left conjugacy closed loops are obtained from right conjugacy closed loops Left conjugacy closed loops are obtained from right conjugacy closed loops
via Opposite. via Opposite.
The library named CC contains all nonassociative conjugacy closed loops of The library named CC contains all CC loops of order 2le 2^kle 64, 3le 3^kle
order nle 27 and also of orders 2p and p^2 for all primes p. 81, 5le 5^kle 125, 7le 7^kle 343, all nonassociative CC loops of order less
than 28, and all nonassociative CC loops of order p^2 and 2p for any odd
prime p.
By results of Kunen [Kun00], for every odd prime p there are precisely 3 By results of Kunen [Kun00], for every odd prime p there are precisely 3
nonassociative conjugacy closed loops of order p^2. Csörgő and Drápal [CD05] nonassociative conjugacy closed loops of order p^2. Csörgő and Drápal [CD05]
@ -215,25 +239,25 @@
m + n ). m + n ).
9.6-3 CCLoop and ConjugacyClosedLoop 9.7-3 CCLoop and ConjugacyClosedLoop
CCLoop( n, m )  function CCLoop( n, m )  function
ConjugacyClosedLoop( n, m )  function ConjugacyClosedLoop( n, m )  function
Returns: The mth conjugacy closed loop of order n in the library. Returns: The mth conjugacy closed loop of order n in the library.
9.7 Small Loops 9.8 Small Loops
The library named small contains all nonassociative loops of order 5 and 6. The library named small contains all nonassociative loops of order 5 and 6.
There are 5 and 107 such loops, respectively. There are 5 and 107 such loops, respectively.
9.7-1 SmallLoop 9.8-1 SmallLoop
SmallLoop( n, m )  function SmallLoop( n, m )  function
Returns: The mth loop of order n in the library. Returns: The mth loop of order n in the library.
9.8 Paige Loops 9.9 Paige Loops
Paige loops are nonassociative finite simple Moufang loops. By [Lie87], Paige loops are nonassociative finite simple Moufang loops. By [Lie87],
there is precisely one Paige loop for every finite field. there is precisely one Paige loop for every finite field.
@ -241,14 +265,14 @@
The library named Paige contains the smallest nonassociative simple Moufang The library named Paige contains the smallest nonassociative simple Moufang
loop. loop.
9.8-1 PaigeLoop 9.9-1 PaigeLoop
PaigeLoop( q )  function PaigeLoop( q )  function
Returns: The Paige loop constructed over the finite field of order q. Only Returns: The Paige loop constructed over the finite field of order q. Only
the case q=2 is implemented. the case q=2 is implemented.
9.9 Nilpotent Loops 9.10 Nilpotent Loops
The library named nilpotent contains all nonassociative nilpotent loops of The library named nilpotent contains all nonassociative nilpotent loops of
order less than 12 up to isomorphism. There are 2 nonassociative nilpotent order less than 12 up to isomorphism. There are 2 nonassociative nilpotent
@ -258,30 +282,32 @@
are 2623755 nilpotent loops of order 12, and 123794003928541545927226368 are 2623755 nilpotent loops of order 12, and 123794003928541545927226368
nilpotent loops of order 22. nilpotent loops of order 22.
9.9-1 NilpotentLoop 9.10-1 NilpotentLoop
NilpotentLoop( n, m )  function NilpotentLoop( n, m )  function
Returns: The mth nilpotent loop of order n in the library. Returns: The mth nilpotent loop of order n in the library.
9.10 Automorphic Loops 9.11 Automorphic Loops
The library named automorphic contains all nonassociative automorphic loops The library named automorphic contains all nonassociative automorphic loops
of order less than 16 up to isomorphism (there is 1 such loop of order 6, 7 of order less than 16 up to isomorphism (there is 1 such loop of order 6, 7
of order 8, 3 of order 10, 2 of order 12, 5 of order 14, and 2 of order 15), of order 8, 3 of order 10, 2 of order 12, 5 of order 14, and 2 of order 15)
all commutative automorphic loops of order 3, 9, 27 and 81 (there are 1, 2, and all commutative automorphic loops of order 3, 9, 27 and 81 (there are 1,
7 and 72 such loops, respectively, including abelian groups), and 2, 7 and 72 such loops).
commutative automorphic loops Q of order 243 possessing a central subloop S
of order 3 such that Q/S is not the elementary abelian group of order 81
(there are 118451 such loops).
9.10-1 AutomorphicLoop It turns out that commutative automorphic loops of order 3, 9, 27 and 81
(but not 243) are in one-to-on correspondence with left Bruck loops of the
respective orders, see [Gre14], [SV17]. Only the left Bruck loops are stored
in the library.
9.11-1 AutomorphicLoop
AutomorphicLoop( n, m )  function AutomorphicLoop( n, m )  function
Returns: The mth automorphic loop of order n in the library. Returns: The mth automorphic loop of order n in the library.
9.11 Interesting Loops 9.12 Interesting Loops
The library named interesting contains some loops that are illustrative in The library named interesting contains some loops that are illustrative in
the theory of loops. At this point, the library contains a nonassociative the theory of loops. At this point, the library contains a nonassociative
@ -290,20 +316,20 @@
generalize octonions), and the unique nonassociative simple right Bol loop generalize octonions), and the unique nonassociative simple right Bol loop
of order 96 and exponent 2. of order 96 and exponent 2.
9.11-1 InterestingLoop 9.12-1 InterestingLoop
InterestingLoop( n, m )  function InterestingLoop( n, m )  function
Returns: The mth interesting loop of order n in the library. Returns: The mth interesting loop of order n in the library.
9.12 Libraries of Loops Up To Isotopism 9.13 Libraries of Loops Up To Isotopism
For the library named small we also provide the corresponding library of For the library named small we also provide the corresponding library of
loops up to isotopism. In general, given a library named libname, the loops up to isotopism. In general, given a library named libname, the
corresponding library of loops up to isotopism is named itp lib, and the corresponding library of loops up to isotopism is named itp lib, and the
loops can be retrieved by the template ItpLibLoop(n,m). loops can be retrieved by the template ItpLibLoop(n,m).
9.12-1 ItpSmallLoop 9.13-1 ItpSmallLoop
ItpSmallLoop( n, m )  function ItpSmallLoop( n, m )  function
Returns: The mth small loop of order n up to isotopism in the library. Returns: The mth small loop of order n up to isotopism in the library.

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@ -38,60 +38,66 @@
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X7EE99F647C537994">9.2-1 LeftBolLoop</a></span> <span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X7EE99F647C537994">9.2-1 LeftBolLoop</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X8774304282654C58">9.2-2 RightBolLoop</a></span> <span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X8774304282654C58">9.2-2 RightBolLoop</a></span>
</div></div> </div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X7953702D84E60AF4">9.3 <span class="Heading">Moufang Loops</span></a> <div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X8028D69A86B15897">9.3 <span class="Heading">Left Bruck Loops and Right Bruck Loops</span></a>
</span> </span>
<div class="ContSSBlock"> <div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X81E82098822543EE">9.3-1 MoufangLoop</a></span> <span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X8290B01780F0FCD3">9.3-1 LeftBruckLoop</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X798DD7CF871F648F">9.3-2 RightBruckLoop</a></span>
</div></div> </div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X7BCA6BCB847F79DC">9.4 <span class="Heading">Code Loops</span></a> <div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X7953702D84E60AF4">9.4 <span class="Heading">Moufang Loops</span></a>
</span> </span>
<div class="ContSSBlock"> <div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X7DB4D3B27BB4D7EE">9.4-1 CodeLoop</a></span> <span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X81E82098822543EE">9.4-1 MoufangLoop</a></span>
</div></div> </div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X84E941EE7846D3EE">9.5 <span class="Heading">Steiner Loops</span></a> <div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X7BCA6BCB847F79DC">9.5 <span class="Heading">Code Loops</span></a>
</span> </span>
<div class="ContSSBlock"> <div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X87C235457E859AF4">9.5-1 SteinerLoop</a></span> <span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X7DB4D3B27BB4D7EE">9.5-1 CodeLoop</a></span>
</div></div> </div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X867E5F0783FEB8B5">9.6 <span class="Heading">Conjugacy Closed Loops</span></a> <div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X84E941EE7846D3EE">9.6 <span class="Heading">Steiner Loops</span></a>
</span> </span>
<div class="ContSSBlock"> <div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X806B2DE67990E42F">9.6-1 <span class="Heading">RCCLoop and RightConjugacyClosedLoop</span></a> <span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X87C235457E859AF4">9.6-1 SteinerLoop</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X867E5F0783FEB8B5">9.7 <span class="Heading">Conjugacy Closed Loops</span></a>
</span> </span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X80AB8B107D55FB19">9.6-2 <span class="Heading">LCCLoop and LeftConjugacyClosedLoop</span></a> <div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X806B2DE67990E42F">9.7-1 <span class="Heading">RCCLoop and RightConjugacyClosedLoop</span></a>
</span> </span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X798BC601843E8916">9.6-3 <span class="Heading">CCLoop and ConjugacyClosedLoop</span></a> <span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X80AB8B107D55FB19">9.7-2 <span class="Heading">LCCLoop and LeftConjugacyClosedLoop</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X798BC601843E8916">9.7-3 <span class="Heading">CCLoop and ConjugacyClosedLoop</span></a>
</span> </span>
</div></div> </div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X7E3A8F2C790F2CA1">9.7 <span class="Heading">Small Loops</span></a> <div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X7E3A8F2C790F2CA1">9.8 <span class="Heading">Small Loops</span></a>
</span> </span>
<div class="ContSSBlock"> <div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X7C6EE23E84CD87D3">9.7-1 SmallLoop</a></span> <span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X7C6EE23E84CD87D3">9.8-1 SmallLoop</a></span>
</div></div> </div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X8135C8FD8714C606">9.8 <span class="Heading">Paige Loops</span></a> <div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X8135C8FD8714C606">9.9 <span class="Heading">Paige Loops</span></a>
</span> </span>
<div class="ContSSBlock"> <div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X7FCF4D6B7AD66D74">9.8-1 PaigeLoop</a></span> <span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X7FCF4D6B7AD66D74">9.9-1 PaigeLoop</a></span>
</div></div> </div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X86695C577A4D1784">9.9 <span class="Heading">Nilpotent Loops</span></a> <div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X86695C577A4D1784">9.10 <span class="Heading">Nilpotent Loops</span></a>
</span> </span>
<div class="ContSSBlock"> <div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X7A9C960D86E2AD28">9.9-1 NilpotentLoop</a></span> <span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X7A9C960D86E2AD28">9.10-1 NilpotentLoop</a></span>
</div></div> </div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X793B22EA8643C667">9.10 <span class="Heading">Automorphic Loops</span></a> <div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X793B22EA8643C667">9.11 <span class="Heading">Automorphic Loops</span></a>
</span> </span>
<div class="ContSSBlock"> <div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X784FFA9E7FDA9F43">9.10-1 AutomorphicLoop</a></span> <span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X784FFA9E7FDA9F43">9.11-1 AutomorphicLoop</a></span>
</div></div> </div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X843BD73F788049F7">9.11 <span class="Heading">Interesting Loops</span></a> <div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X843BD73F788049F7">9.12 <span class="Heading">Interesting Loops</span></a>
</span> </span>
<div class="ContSSBlock"> <div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X87F24AD3811910D3">9.11-1 InterestingLoop</a></span> <span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X87F24AD3811910D3">9.12-1 InterestingLoop</a></span>
</div></div> </div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X864839227D5C0A90">9.12 <span class="Heading">Libraries of Loops Up To Isotopism</span></a> <div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X864839227D5C0A90">9.13 <span class="Heading">Libraries of Loops Up To Isotopism</span></a>
</span> </span>
<div class="ContSSBlock"> <div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X850C4C01817A098D">9.12-1 ItpSmallLoop</a></span> <span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X850C4C01817A098D">9.13-1 ItpSmallLoop</a></span>
</div></div> </div></div>
</div> </div>
@ -170,15 +176,37 @@
<p><strong class="button">Remark:</strong> Only left Bol loops are stored in the library. Right Bol loops are retrieved by calling <code class="code">Opposite</code> on left Bol loops.</p> <p><strong class="button">Remark:</strong> Only left Bol loops are stored in the library. Right Bol loops are retrieved by calling <code class="code">Opposite</code> on left Bol loops.</p>
<p><a id="X8028D69A86B15897" name="X8028D69A86B15897"></a></p>
<h4>9.3 <span class="Heading">Left Bruck Loops and Right Bruck Loops</span></h4>
<p>The emmerging library named <em>left Bruck</em> contains all left Bruck loops of orders <span class="SimpleMath">\(3\)</span>, <span class="SimpleMath">\(9\)</span>, <span class="SimpleMath">\(27\)</span> and <span class="SimpleMath">\(81\)</span> (there are <span class="SimpleMath">\(1\)</span>, <span class="SimpleMath">\(2\)</span>, <span class="SimpleMath">\(7\)</span> and <span class="SimpleMath">\(72\)</span> such loops, respectively).</p>
<p>For an odd prime <span class="SimpleMath">\(p\)</span>, left Bruck loops of order <span class="SimpleMath">\(p^k\)</span> are centrally nilpotent and hence central extensions of the cyclic group of order <span class="SimpleMath">\(p\)</span> by a left Bruck loop of order <span class="SimpleMath">\(p^{k-1}\)</span>. It is known that left Bruck loops of order <span class="SimpleMath">\(p\)</span> and <span class="SimpleMath">\(p^2\)</span> are abelian groups; we have included them in the library because of the iterative nature of the construction of nilpotent loops.</p>
<p><a id="X8290B01780F0FCD3" name="X8290B01780F0FCD3"></a></p>
<h5>9.3-1 LeftBruckLoop</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; LeftBruckLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: The <var class="Arg">m</var>th left Bruck loop of order <var class="Arg">n</var> in the library.</p>
<p><a id="X798DD7CF871F648F" name="X798DD7CF871F648F"></a></p>
<h5>9.3-2 RightBruckLoop</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; RightBruckLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: The <var class="Arg">m</var>th right Bruck loop of order <var class="Arg">n</var> in the library.</p>
<p><a id="X7953702D84E60AF4" name="X7953702D84E60AF4"></a></p> <p><a id="X7953702D84E60AF4" name="X7953702D84E60AF4"></a></p>
<h4>9.3 <span class="Heading">Moufang Loops</span></h4> <h4>9.4 <span class="Heading">Moufang Loops</span></h4>
<p>The library named <em>Moufang</em> contains all nonassociative Moufang loops of order <span class="SimpleMath">\(n\le 64\)</span> and <span class="SimpleMath">\(n\in\{81,243\}\)</span>.</p> <p>The library named <em>Moufang</em> contains all nonassociative Moufang loops of order <span class="SimpleMath">\(n\le 64\)</span> and <span class="SimpleMath">\(n\in\{81,243\}\)</span>.</p>
<p><a id="X81E82098822543EE" name="X81E82098822543EE"></a></p> <p><a id="X81E82098822543EE" name="X81E82098822543EE"></a></p>
<h5>9.3-1 MoufangLoop</h5> <h5>9.4-1 MoufangLoop</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; MoufangLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; MoufangLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: The <var class="Arg">m</var>th Moufang loop of order <var class="Arg">n</var> in the library.</p> <p>Returns: The <var class="Arg">m</var>th Moufang loop of order <var class="Arg">n</var> in the library.</p>
@ -198,26 +226,26 @@
<p><a id="X7BCA6BCB847F79DC" name="X7BCA6BCB847F79DC"></a></p> <p><a id="X7BCA6BCB847F79DC" name="X7BCA6BCB847F79DC"></a></p>
<h4>9.4 <span class="Heading">Code Loops</span></h4> <h4>9.5 <span class="Heading">Code Loops</span></h4>
<p>The library named <em>code</em> contains all nonassociative code loops of order less than 65. There are 5 such loops of order 16, 16 of order 32, and 80 of order 64, all Moufang. The library merely points to the corresponding Moufang loops. See <a href="chapBib_mj.html#biBNaVo2007">[NV07]</a> for a classification of small code loops.</p> <p>The library named <em>code</em> contains all nonassociative code loops of order less than 65. There are 5 such loops of order 16, 16 of order 32, and 80 of order 64, all Moufang. The library merely points to the corresponding Moufang loops. See <a href="chapBib_mj.html#biBNaVo2007">[NV07]</a> for a classification of small code loops.</p>
<p><a id="X7DB4D3B27BB4D7EE" name="X7DB4D3B27BB4D7EE"></a></p> <p><a id="X7DB4D3B27BB4D7EE" name="X7DB4D3B27BB4D7EE"></a></p>
<h5>9.4-1 CodeLoop</h5> <h5>9.5-1 CodeLoop</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; CodeLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; CodeLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: The <var class="Arg">m</var>th code loop of order <var class="Arg">n</var> in the library.</p> <p>Returns: The <var class="Arg">m</var>th code loop of order <var class="Arg">n</var> in the library.</p>
<p><a id="X84E941EE7846D3EE" name="X84E941EE7846D3EE"></a></p> <p><a id="X84E941EE7846D3EE" name="X84E941EE7846D3EE"></a></p>
<h4>9.5 <span class="Heading">Steiner Loops</span></h4> <h4>9.6 <span class="Heading">Steiner Loops</span></h4>
<p>Here is how the libary named <em>Steiner</em> is described within <strong class="pkg">LOOPS</strong>:</p> <p>Here is how the libary named <em>Steiner</em> is described within <strong class="pkg">LOOPS</strong>:</p>
<div class="example"><pre> <div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">DisplayLibraryInfo( "Steiner" ); </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">DisplayLibraryInfo( "Steiner" );</span>
The library contains all nonassociative Steiner loops of order less or equal to 16. The library contains all nonassociative Steiner loops of order less or equal to 16.
It also contains the associative Steiner loops of order 4 and 8. It also contains the associative Steiner loops of order 4 and 8.
------ ------
@ -234,14 +262,14 @@ true
<p><a id="X87C235457E859AF4" name="X87C235457E859AF4"></a></p> <p><a id="X87C235457E859AF4" name="X87C235457E859AF4"></a></p>
<h5>9.5-1 SteinerLoop</h5> <h5>9.6-1 SteinerLoop</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; SteinerLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; SteinerLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: The <var class="Arg">m</var>th Steiner loop of order <var class="Arg">n</var> in the library.</p> <p>Returns: The <var class="Arg">m</var>th Steiner loop of order <var class="Arg">n</var> in the library.</p>
<p><a id="X867E5F0783FEB8B5" name="X867E5F0783FEB8B5"></a></p> <p><a id="X867E5F0783FEB8B5" name="X867E5F0783FEB8B5"></a></p>
<h4>9.6 <span class="Heading">Conjugacy Closed Loops</span></h4> <h4>9.7 <span class="Heading">Conjugacy Closed Loops</span></h4>
<p>The library named <em>RCC</em> contains all nonassocitive right conjugacy closed loops of order <span class="SimpleMath">\(n\le 27\)</span> up to isomorphism. The data for the library was generated by Katharina Artic <a href="chapBib_mj.html#biBArtic">[Art15]</a> who can also provide additional data for all right conjugacy closed loops of order <span class="SimpleMath">\(n\le 31\)</span>.</p> <p>The library named <em>RCC</em> contains all nonassocitive right conjugacy closed loops of order <span class="SimpleMath">\(n\le 27\)</span> up to isomorphism. The data for the library was generated by Katharina Artic <a href="chapBib_mj.html#biBArtic">[Art15]</a> who can also provide additional data for all right conjugacy closed loops of order <span class="SimpleMath">\(n\le 31\)</span>.</p>
@ -259,7 +287,7 @@ true
<p><a id="X806B2DE67990E42F" name="X806B2DE67990E42F"></a></p> <p><a id="X806B2DE67990E42F" name="X806B2DE67990E42F"></a></p>
<h5>9.6-1 <span class="Heading">RCCLoop and RightConjugacyClosedLoop</span></h5> <h5>9.7-1 <span class="Heading">RCCLoop and RightConjugacyClosedLoop</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; RCCLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; RCCLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; RightConjugacyClosedLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; RightConjugacyClosedLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
@ -267,7 +295,7 @@ true
<p><a id="X80AB8B107D55FB19" name="X80AB8B107D55FB19"></a></p> <p><a id="X80AB8B107D55FB19" name="X80AB8B107D55FB19"></a></p>
<h5>9.6-2 <span class="Heading">LCCLoop and LeftConjugacyClosedLoop</span></h5> <h5>9.7-2 <span class="Heading">LCCLoop and LeftConjugacyClosedLoop</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; LCCLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; LCCLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; LeftConjugacyClosedLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; LeftConjugacyClosedLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
@ -275,7 +303,7 @@ true
<p><strong class="button">Remark:</strong> Only the right conjugacy closed loops are stored in the library. Left conjugacy closed loops are obtained from right conjugacy closed loops via <code class="code">Opposite</code>.<br /></p> <p><strong class="button">Remark:</strong> Only the right conjugacy closed loops are stored in the library. Left conjugacy closed loops are obtained from right conjugacy closed loops via <code class="code">Opposite</code>.<br /></p>
<p>The library named <em>CC</em> contains all nonassociative conjugacy closed loops of order <span class="SimpleMath">\(n\le 27\)</span> and also of orders <span class="SimpleMath">\(2p\)</span> and <span class="SimpleMath">\(p^2\)</span> for all primes <span class="SimpleMath">\(p\)</span>.</p> <p>The library named <em>CC</em> contains all CC loops of order <span class="SimpleMath">\(2\le 2^k\le 64\)</span>, <span class="SimpleMath">\(3\le 3^k\le 81\)</span>, <span class="SimpleMath">\(5\le 5^k\le 125\)</span>, <span class="SimpleMath">\(7\le 7^k\le 343\)</span>, all nonassociative CC loops of order less than 28, and all nonassociative CC loops of order <span class="SimpleMath">\(p^2\)</span> and <span class="SimpleMath">\(2p\)</span> for any odd prime <span class="SimpleMath">\(p\)</span>.</p>
<p>By results of Kunen <a href="chapBib_mj.html#biBKun">[Kun00]</a>, for every odd prime <span class="SimpleMath">\(p\)</span> there are precisely 3 nonassociative conjugacy closed loops of order <span class="SimpleMath">\(p^2\)</span>. Csörgő and Drápal <a href="chapBib_mj.html#biBCsDr">[CD05]</a> described these 3 loops by multiplicative formulas on <span class="SimpleMath">\(\mathbb{Z}_{p^2}\)</span> and <span class="SimpleMath">\(\mathbb{Z}_p \times \mathbb{Z}_p\)</span> as follows:</p> <p>By results of Kunen <a href="chapBib_mj.html#biBKun">[Kun00]</a>, for every odd prime <span class="SimpleMath">\(p\)</span> there are precisely 3 nonassociative conjugacy closed loops of order <span class="SimpleMath">\(p^2\)</span>. Csörgő and Drápal <a href="chapBib_mj.html#biBCsDr">[CD05]</a> described these 3 loops by multiplicative formulas on <span class="SimpleMath">\(\mathbb{Z}_{p^2}\)</span> and <span class="SimpleMath">\(\mathbb{Z}_p \times \mathbb{Z}_p\)</span> as follows:</p>
@ -295,7 +323,7 @@ true
<p><a id="X798BC601843E8916" name="X798BC601843E8916"></a></p> <p><a id="X798BC601843E8916" name="X798BC601843E8916"></a></p>
<h5>9.6-3 <span class="Heading">CCLoop and ConjugacyClosedLoop</span></h5> <h5>9.7-3 <span class="Heading">CCLoop and ConjugacyClosedLoop</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; CCLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; CCLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ConjugacyClosedLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ConjugacyClosedLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
@ -303,20 +331,20 @@ true
<p><a id="X7E3A8F2C790F2CA1" name="X7E3A8F2C790F2CA1"></a></p> <p><a id="X7E3A8F2C790F2CA1" name="X7E3A8F2C790F2CA1"></a></p>
<h4>9.7 <span class="Heading">Small Loops</span></h4> <h4>9.8 <span class="Heading">Small Loops</span></h4>
<p>The library named <em>small</em> contains all nonassociative loops of order 5 and 6. There are 5 and 107 such loops, respectively.</p> <p>The library named <em>small</em> contains all nonassociative loops of order 5 and 6. There are 5 and 107 such loops, respectively.</p>
<p><a id="X7C6EE23E84CD87D3" name="X7C6EE23E84CD87D3"></a></p> <p><a id="X7C6EE23E84CD87D3" name="X7C6EE23E84CD87D3"></a></p>
<h5>9.7-1 SmallLoop</h5> <h5>9.8-1 SmallLoop</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; SmallLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; SmallLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: The <var class="Arg">m</var>th loop of order <var class="Arg">n</var> in the library.</p> <p>Returns: The <var class="Arg">m</var>th loop of order <var class="Arg">n</var> in the library.</p>
<p><a id="X8135C8FD8714C606" name="X8135C8FD8714C606"></a></p> <p><a id="X8135C8FD8714C606" name="X8135C8FD8714C606"></a></p>
<h4>9.8 <span class="Heading">Paige Loops</span></h4> <h4>9.9 <span class="Heading">Paige Loops</span></h4>
<p><em>Paige loops</em> are nonassociative finite simple Moufang loops. By <a href="chapBib_mj.html#biBLi">[Lie87]</a>, there is precisely one Paige loop for every finite field.</p> <p><em>Paige loops</em> are nonassociative finite simple Moufang loops. By <a href="chapBib_mj.html#biBLi">[Lie87]</a>, there is precisely one Paige loop for every finite field.</p>
@ -324,14 +352,14 @@ true
<p><a id="X7FCF4D6B7AD66D74" name="X7FCF4D6B7AD66D74"></a></p> <p><a id="X7FCF4D6B7AD66D74" name="X7FCF4D6B7AD66D74"></a></p>
<h5>9.8-1 PaigeLoop</h5> <h5>9.9-1 PaigeLoop</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; PaigeLoop</code>( <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; PaigeLoop</code>( <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: The Paige loop constructed over the finite field of order <var class="Arg">q</var>. Only the case <code class="code"><var class="Arg">q</var>=2</code> is implemented.</p> <p>Returns: The Paige loop constructed over the finite field of order <var class="Arg">q</var>. Only the case <code class="code"><var class="Arg">q</var>=2</code> is implemented.</p>
<p><a id="X86695C577A4D1784" name="X86695C577A4D1784"></a></p> <p><a id="X86695C577A4D1784" name="X86695C577A4D1784"></a></p>
<h4>9.9 <span class="Heading">Nilpotent Loops</span></h4> <h4>9.10 <span class="Heading">Nilpotent Loops</span></h4>
<p>The library named <em>nilpotent</em> contains all nonassociative nilpotent loops of order less than 12 up to isomorphism. There are 2 nonassociative nilpotent loops of order 6, 134 of order 8, 8 of order 9 and 1043 of order 10.</p> <p>The library named <em>nilpotent</em> contains all nonassociative nilpotent loops of order less than 12 up to isomorphism. There are 2 nonassociative nilpotent loops of order 6, 134 of order 8, 8 of order 9 and 1043 of order 10.</p>
@ -339,57 +367,59 @@ true
<p><a id="X7A9C960D86E2AD28" name="X7A9C960D86E2AD28"></a></p> <p><a id="X7A9C960D86E2AD28" name="X7A9C960D86E2AD28"></a></p>
<h5>9.9-1 NilpotentLoop</h5> <h5>9.10-1 NilpotentLoop</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; NilpotentLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; NilpotentLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: The <var class="Arg">m</var>th nilpotent loop of order <var class="Arg">n</var> in the library.</p> <p>Returns: The <var class="Arg">m</var>th nilpotent loop of order <var class="Arg">n</var> in the library.</p>
<p><a id="X793B22EA8643C667" name="X793B22EA8643C667"></a></p> <p><a id="X793B22EA8643C667" name="X793B22EA8643C667"></a></p>
<h4>9.10 <span class="Heading">Automorphic Loops</span></h4> <h4>9.11 <span class="Heading">Automorphic Loops</span></h4>
<p>The library named <em>automorphic</em> contains all nonassociative automorphic loops of order less than 16 up to isomorphism (there is 1 such loop of order 6, 7 of order 8, 3 of order 10, 2 of order 12, 5 of order 14, and 2 of order 15), all commutative automorphic loops of order 3, 9, 27 and 81 (there are 1, 2, 7 and 72 such loops, respectively, including abelian groups), and commutative automorphic loops <span class="SimpleMath">\(Q\)</span> of order 243 possessing a central subloop <span class="SimpleMath">\(S\)</span> of order 3 such that <span class="SimpleMath">\(Q/S\)</span> is not the elementary abelian group of order 81 (there are 118451 such loops).</p> <p>The library named <em>automorphic</em> contains all nonassociative automorphic loops of order less than 16 up to isomorphism (there is 1 such loop of order 6, 7 of order 8, 3 of order 10, 2 of order 12, 5 of order 14, and 2 of order 15) and all commutative automorphic loops of order 3, 9, 27 and 81 (there are 1, 2, 7 and 72 such loops).</p>
<p>It turns out that commutative automorphic loops of order 3, 9, 27 and 81 (but not 243) are in one-to-on correspondence with left Bruck loops of the respective orders, see <a href="chapBib_mj.html#biBGreer">[Gre14]</a>, <a href="chapBib_mj.html#biBStuhlVojtechovsky">[SV17]</a>. Only the left Bruck loops are stored in the library.</p>
<p><a id="X784FFA9E7FDA9F43" name="X784FFA9E7FDA9F43"></a></p> <p><a id="X784FFA9E7FDA9F43" name="X784FFA9E7FDA9F43"></a></p>
<h5>9.10-1 AutomorphicLoop</h5> <h5>9.11-1 AutomorphicLoop</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; AutomorphicLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; AutomorphicLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: The <var class="Arg">m</var>th automorphic loop of order <var class="Arg">n</var> in the library.</p> <p>Returns: The <var class="Arg">m</var>th automorphic loop of order <var class="Arg">n</var> in the library.</p>
<p><a id="X843BD73F788049F7" name="X843BD73F788049F7"></a></p> <p><a id="X843BD73F788049F7" name="X843BD73F788049F7"></a></p>
<h4>9.11 <span class="Heading">Interesting Loops</span></h4> <h4>9.12 <span class="Heading">Interesting Loops</span></h4>
<p>The library named <em>interesting</em> contains some loops that are illustrative in the theory of loops. At this point, the library contains a nonassociative loop of order 5, a nonassociative nilpotent loop of order 6, a non-Moufang left Bol loop of order 16, the loop of sedenions of order 32 (sedenions generalize octonions), and the unique nonassociative simple right Bol loop of order 96 and exponent 2.</p> <p>The library named <em>interesting</em> contains some loops that are illustrative in the theory of loops. At this point, the library contains a nonassociative loop of order 5, a nonassociative nilpotent loop of order 6, a non-Moufang left Bol loop of order 16, the loop of sedenions of order 32 (sedenions generalize octonions), and the unique nonassociative simple right Bol loop of order 96 and exponent 2.</p>
<p><a id="X87F24AD3811910D3" name="X87F24AD3811910D3"></a></p> <p><a id="X87F24AD3811910D3" name="X87F24AD3811910D3"></a></p>
<h5>9.11-1 InterestingLoop</h5> <h5>9.12-1 InterestingLoop</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; InterestingLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; InterestingLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: The <var class="Arg">m</var>th interesting loop of order <var class="Arg">n</var> in the library.</p> <p>Returns: The <var class="Arg">m</var>th interesting loop of order <var class="Arg">n</var> in the library.</p>
<p><a id="X864839227D5C0A90" name="X864839227D5C0A90"></a></p> <p><a id="X864839227D5C0A90" name="X864839227D5C0A90"></a></p>
<h4>9.12 <span class="Heading">Libraries of Loops Up To Isotopism</span></h4> <h4>9.13 <span class="Heading">Libraries of Loops Up To Isotopism</span></h4>
<p>For the library named <em>small</em> we also provide the corresponding library of loops up to isotopism. In general, given a library named <em>libname</em>, the corresponding library of loops up to isotopism is named <em>itp lib</em>, and the loops can be retrieved by the template <code class="code">ItpLibLoop(n,m)</code>.</p> <p>For the library named <em>small</em> we also provide the corresponding library of loops up to isotopism. In general, given a library named <em>libname</em>, the corresponding library of loops up to isotopism is named <em>itp lib</em>, and the loops can be retrieved by the template <code class="code">ItpLibLoop(n,m)</code>.</p>
<p><a id="X850C4C01817A098D" name="X850C4C01817A098D"></a></p> <p><a id="X850C4C01817A098D" name="X850C4C01817A098D"></a></p>
<h5>9.12-1 ItpSmallLoop</h5> <h5>9.13-1 ItpSmallLoop</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ItpSmallLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ItpSmallLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: The <var class="Arg">m</var>th small loop of order <var class="Arg">n</var> up to isotopism in the library.</p> <p>Returns: The <var class="Arg">m</var>th small loop of order <var class="Arg">n</var> up to isotopism in the library.</p>
<div class="example"><pre> <div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">SmallLoop( 6, 14 ); </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">SmallLoop( 6, 14 );</span>
&lt;small loop 6/14&gt; &lt;small loop 6/14&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">ItpSmallLoop( 6, 14 ); </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">ItpSmallLoop( 6, 14 );</span>
&lt;small loop 6/42&gt; &lt;small loop 6/42&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LibraryLoop( "itp small", 6, 14 ); </span> <span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LibraryLoop( "itp small", 6, 14 );</span>
&lt;small loop 6/42&gt; &lt;small loop 6/42&gt;
</pre></div> </pre></div>

7
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@ -105,9 +105,6 @@
( IsLeftAutomorphicLoop, IsAutomorphicLoop ) ( IsLeftAutomorphicLoop, IsAutomorphicLoop )
( IsRightAutomorphicLoop, IsAutomorphicLoop ) ( IsRightAutomorphicLoop, IsAutomorphicLoop )
( IsMiddleAutomorphicLoop, IsAutomorphicLoop ) ( IsMiddleAutomorphicLoop, IsAutomorphicLoop )
( IsMiddleAutomorphicLoop, IsCommutative )
( IsAutomorphicLoop, IsLeftAutomorphicLoop and IsCommutative )
( IsAutomorphicLoop, IsRightAutomorphicLoop and IsCommutative )
( IsLeftAutomorphicLoop, IsRightAutomorphicLoop and ( IsLeftAutomorphicLoop, IsRightAutomorphicLoop and
HasAntiautomorphicInverseProperty ) HasAntiautomorphicInverseProperty )
( IsRightAutomorphicLoop, IsLeftAutomorphicLoop and ( IsRightAutomorphicLoop, IsLeftAutomorphicLoop and
@ -120,9 +117,13 @@
( IsMoufangLoop, IsAutomorphicLoop and HasLeftInverseProperty ) ( IsMoufangLoop, IsAutomorphicLoop and HasLeftInverseProperty )
( IsMoufangLoop, IsAutomorphicLoop and HasRightInverseProperty ) ( IsMoufangLoop, IsAutomorphicLoop and HasRightInverseProperty )
( IsMoufangLoop, IsAutomorphicLoop and HasWeakInverseProperty ) ( IsMoufangLoop, IsAutomorphicLoop and HasWeakInverseProperty )
( IsMiddleAutomorphicLoop, IsCommutative )
( IsLeftAutomorphicLoop, IsLeftBruckLoop ) ( IsLeftAutomorphicLoop, IsLeftBruckLoop )
( IsLeftAutomorphicLoop, IsLCCLoop ) ( IsLeftAutomorphicLoop, IsLCCLoop )
( IsRightAutomorphicLoop, IsRightBruckLoop ) ( IsRightAutomorphicLoop, IsRightBruckLoop )
( IsRightAutomorphicLoop, IsRCCLoop ) ( IsRightAutomorphicLoop, IsRCCLoop )
( IsAutomorphicLoop, IsCommutative and IsMoufangLoop ) ( IsAutomorphicLoop, IsCommutative and IsMoufangLoop )
( IsAutomorphicLoop, IsLeftAutomorphicLoop and IsMiddleAutomorphicLoop )
( IsAutomorphicLoop, IsRightAutomorphicLoop and IsMiddleAutomorphicLoop )
( IsAutomorphicLoop, IsAssociative )

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@ -42,6 +42,9 @@
less than 64, Nova Science Publishers Inc., Commack, NY (1999), xviii+287 less than 64, Nova Science Publishers Inc., Commack, NY (1999), xviii+287
pages. pages.
[Gre14] Greer, M., A class of loops categorically isomorphic to Bruck loops
of odd order, Comm. Algebra, 42, 8 (2014), 3682–3697.
[GKN14] Grishkov, A., Kinyon, M. and Nagy, G. P., Solvability of commutative [GKN14] Grishkov, A., Kinyon, M. and Nagy, G. P., Solvability of commutative
automorphic loops, Proc. Amer. Math. Soc., 142, 9 (2014), 3029–3037. automorphic loops, Proc. Amer. Math. Soc., 142, 9 (2014), 3029–3037.
@ -89,6 +92,10 @@
[SZ12] Slattery, M. and Zenisek, A., Moufang loops of order 243, [SZ12] Slattery, M. and Zenisek, A., Moufang loops of order 243,
Commentationes Mathematicae Universitatis Carolinae, 53, 3 (2012), 423–428. Commentationes Mathematicae Universitatis Carolinae, 53, 3 (2012), 423–428.
[SV17] Stuhl, I. and Vojtěchovský, P., Involutory latin quandles, Bruck
loops and commutative automorphic loops of odd prime power order,  (2017),
((preprint)).
[Voj06] Vojtěchovský, P., Toward the classification of Moufang loops of [Voj06] Vojtěchovský, P., Toward the classification of Moufang loops of
order 64, European J. Combin., 27, 3 (2006), 444–460. order 64, European J. Combin., 27, 3 (2006), 444–460.

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@ -164,6 +164,18 @@
</p> </p>
<p><a id="biBGreer" name="biBGreer"></a></p>
<p class='BibEntry'>
[<span class='BibKeyLink'><a href="http://www.ams.org/mathscinet-getitem?mr=3196069">Gre14</a></span>] <b class='BibAuthor'>Greer, M.</b>,
<i class='BibTitle'>A class of loops categorically isomorphic to Bruck loops of
odd order</i>,
<span class='BibJournal'>Comm. Algebra</span>,
<em class='BibVolume'>42</em> (<span class='BibNumber'>8</span>)
(<span class='BibYear'>2014</span>),
<span class='BibPages'>3682–3697</span>.
</p>
<p><a id="biBGrKiNa" name="biBGrKiNa"></a></p> <p><a id="biBGrKiNa" name="biBGrKiNa"></a></p>
<p class='BibEntry'> <p class='BibEntry'>
[<span class='BibKeyLink'><a href="http://www.ams.org/mathscinet-getitem?mr=3223359">GKN14</a></span>] <b class='BibAuthor'>Grishkov, A., Kinyon, M. and Nagy, G. P.</b>, [<span class='BibKeyLink'><a href="http://www.ams.org/mathscinet-getitem?mr=3223359">GKN14</a></span>] <b class='BibAuthor'>Grishkov, A., Kinyon, M. and Nagy, G. P.</b>,
@ -327,6 +339,17 @@
</p> </p>
<p><a id="biBStuhlVojtechovsky" name="biBStuhlVojtechovsky"></a></p>
<p class='BibEntry'>
[<span class='BibKey'>SV17</span>] <b class='BibAuthor'>Stuhl, I. and Vojtěchovský, P.</b>,
<i class='BibTitle'>Involutory latin quandles, Bruck loops and commutative automorphic
loops of odd prime power order</i>,
<span class='BibJournal'></span>
(<span class='BibYear'>2017</span>)<br />
(<span class='BibNote'>preprint</span>).
</p>
<p><a id="biBVo" name="biBVo"></a></p> <p><a id="biBVo" name="biBVo"></a></p>
<p class='BibEntry'> <p class='BibEntry'>
[<span class='BibKeyLink'><a href="http://www.ams.org/mathscinet-getitem?mr=2206479">Voj06</a></span>] <b class='BibAuthor'>Vojtěchovský, P.</b>, [<span class='BibKeyLink'><a href="http://www.ams.org/mathscinet-getitem?mr=2206479">Voj06</a></span>] <b class='BibAuthor'>Vojtěchovský, P.</b>,

54
doc/chapInd.txt
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@ -11,7 +11,7 @@
alternative loop, left 7.4 alternative loop, left 7.4
alternative loop, right 7.4 alternative loop, right 7.4
antiautomorphic inverse property 7.2-5 antiautomorphic inverse property 7.2-5
AreEqualDiscriminators 6.11-9 AreEqualDiscriminators 6.11-11
AssociatedLeftBruckLoop 8.1-1 AssociatedLeftBruckLoop 8.1-1
AssociatedRightBruckLoop 8.1-1 AssociatedRightBruckLoop 8.1-1
associator 2.5 associator 2.5
@ -23,7 +23,7 @@
automorphic loop, left 7.7 automorphic loop, left 7.7
automorphic loop, middle 7.7 automorphic loop, middle 7.7
automorphic loop, right 7.7 automorphic loop, right 7.7
AutomorphicLoop 9.10-1 AutomorphicLoop 9.11-1
AutomorphismGroup 6.11-5 AutomorphismGroup 6.11-5
Bol loop, left 3.3 Bol loop, left 3.3
Bol loop, left 7.4 Bol loop, left 7.4
@ -39,7 +39,7 @@
Cayley table, canonical 4.3-1 Cayley table, canonical 4.3-1
CayleyTable 5.1-2 CayleyTable 5.1-2
CayleyTableByPerms 4.6-1 CayleyTableByPerms 4.6-1
CCLoop 9.6-3 CCLoop 9.7-3
center 2.3 center 2.3
Center 6.6-4 Center 6.6-4
central series, lower 6.9-5 central series, lower 6.9-5
@ -47,7 +47,7 @@
Chein loop 8.2-3 Chein loop 8.2-3
cocycle 4.8 cocycle 4.8
code loop 7.8-1 code loop 7.8-1
CodeLoop 9.4-1 CodeLoop 9.5-1
commutant 2.3 commutant 2.3
Commutant 6.6-3 Commutant 6.6-3
commutator 2.5 commutator 2.5
@ -55,7 +55,7 @@
conjugacy closed loop 7.6 conjugacy closed loop 7.6
conjugacy closed loop, left 7.6 conjugacy closed loop, left 7.6
conjugacy closed loop, right 7.6 conjugacy closed loop, right 7.6
ConjugacyClosedLoop 9.6-3 ConjugacyClosedLoop 9.7-3
conjugation 6.5 conjugation 6.5
coset 6.2-6 coset 6.2-6
derived series 2.4 derived series 2.4
@ -64,7 +64,7 @@
DerivedSubloop 6.10-2 DerivedSubloop 6.10-2
diassociative quasigroup 7.1-4 diassociative quasigroup 7.1-4
DirectProduct 4.11-1 DirectProduct 4.11-1
Discriminator 6.11-8 Discriminator 6.11-10
DisplayLibraryInfo 9.1-3 DisplayLibraryInfo 9.1-3
distributive quasigroup 7.3-6 distributive quasigroup 7.3-6
distributive quasigroup, left 7.3-6 distributive quasigroup, left 7.3-6
@ -111,7 +111,7 @@
inner mapping group, middle 6.5 inner mapping group, middle 6.5
inner mapping group, right 2.2 inner mapping group, right 2.2
InnerMappingGroup 6.5-3 InnerMappingGroup 6.5-3
InterestingLoop 9.11-1 InterestingLoop 9.12-1
IntoGroup 4.10-4 IntoGroup 4.10-4
IntoLoop 4.10-3 IntoLoop 4.10-3
IntoQuasigroup 4.10-1 IntoQuasigroup 4.10-1
@ -169,8 +169,8 @@
IsNilpotent 6.9-1 IsNilpotent 6.9-1
IsNormal 6.7-1 IsNormal 6.7-1
IsNuclearSquareLoop 7.4-11 IsNuclearSquareLoop 7.4-11
IsomorphicCopyByNormalSubloop 6.11-7 IsomorphicCopyByNormalSubloop 6.11-9
IsomorphicCopyByPerm 6.11-6 IsomorphicCopyByPerm 6.11-8
isomorphism 2.6 isomorphism 2.6
IsomorphismLoops 6.11-2 IsomorphismLoops 6.11-2
IsomorphismQuasigroups 6.11-1 IsomorphismQuasigroups 6.11-1
@ -206,16 +206,17 @@
IsSubquasigroup 6.2-3 IsSubquasigroup 6.2-3
IsTotallySymmetric 7.3-2 IsTotallySymmetric 7.3-2
IsUnipotent 7.3-5 IsUnipotent 7.3-5
ItpSmallLoop 9.12-1 ItpSmallLoop 9.13-1
K loop, left 7.8-3 K loop, left 7.8-3
K loop, right 7.8-4 K loop, right 7.8-4
latin square 2.1 latin square 2.1
latin square 4.1 latin square 4.1
latin square, random 4.9 latin square, random 4.9
LC loop 7.4 LC loop 7.4
LCCLoop 9.6-2 LCCLoop 9.7-2
LeftBolLoop 9.2-1 LeftBolLoop 9.2-1
LeftConjugacyClosedLoop 9.6-2 LeftBruckLoop 9.3-1
LeftConjugacyClosedLoop 9.7-2
LeftDivision 5.2-1 LeftDivision 5.2-1
LeftDivision 5.2-1 LeftDivision 5.2-1
LeftDivision 5.2-1 LeftDivision 5.2-1
@ -234,7 +235,7 @@
loop, LC 7.4 loop, LC 7.4
loop, Moufang 7.4 loop, Moufang 7.4
loop, Osborn 7.6-4 loop, Osborn 7.6-4
loop, Paige 9.8 loop, Paige 9.9
loop, RC 7.4 loop, RC 7.4
loop, Steiner 7.8-2 loop, Steiner 7.8-2
loop, alternative 7.4 loop, alternative 7.4
@ -259,7 +260,7 @@
loop, nilpotent 2.4 loop, nilpotent 2.4
loop, nilpotent 4.9-2 loop, nilpotent 4.9-2
loop, nuclear square 7.4 loop, nuclear square 7.4
loop, octonion 9.3-1 loop, octonion 9.4-1
loop, of Bol-Moufang type 7.4 loop, of Bol-Moufang type 7.4
loop, power alternative 7.5 loop, power alternative 7.5
loop, power associative 5.1-5 loop, power associative 5.1-5
@ -271,7 +272,7 @@
loop, right conjugacy closed 7.6 loop, right conjugacy closed 7.6
loop, right nuclear square 7.4 loop, right nuclear square 7.4
loop, right power alternative 7.5 loop, right power alternative 7.5
loop, sedenion 9.11 loop, sedenion 9.12
loop, simple 3.3 loop, simple 3.3
loop, simple 6.7-3 loop, simple 6.7-3
loop, solvable 2.4 loop, solvable 2.4
@ -286,6 +287,7 @@
LoopByRightFolder 4.7-1 LoopByRightFolder 4.7-1
LoopByRightSection 4.6-3 LoopByRightSection 4.6-3
LoopFromFile 4.5-1 LoopFromFile 4.5-1
LoopIsomorph 6.11-7
LoopMG2 8.2-3 LoopMG2 8.2-3
LoopsUpToIsomorphism 6.11-4 LoopsUpToIsomorphism 6.11-4
LoopsUpToIsotopism 6.12-2 LoopsUpToIsotopism 6.12-2
@ -299,7 +301,7 @@
modification, cyclic 8.2-1 modification, cyclic 8.2-1
modification, dihedral 8.2-2 modification, dihedral 8.2-2
Moufang loop 7.4 Moufang loop 7.4
MoufangLoop 9.3-1 MoufangLoop 9.4-1
multiplication group 2.2 multiplication group 2.2
multiplication group, left 2.2 multiplication group, left 2.2
multiplication group, relative 6.4-2 multiplication group, relative 6.4-2
@ -315,7 +317,7 @@
NilpotencyClassOfLoop 6.9-2 NilpotencyClassOfLoop 6.9-2
nilpotent loop 2.4 nilpotent loop 2.4
nilpotent loop, strongly 6.9-3 nilpotent loop, strongly 6.9-3
NilpotentLoop 9.9-1 NilpotentLoop 9.10-1
normal closure 6.7-2 normal closure 6.7-2
normal subloop 6.7-1 normal subloop 6.7-1
NormalClosure 6.7-2 NormalClosure 6.7-2
@ -332,7 +334,7 @@
nucleus, right 2.3 nucleus, right 2.3
NucleusOfLoop 6.6-2 NucleusOfLoop 6.6-2
NucleusOfQuasigroup 6.6-2 NucleusOfQuasigroup 6.6-2
octonion loop 9.3-1 octonion loop 9.4-1
One 5.1-3 One 5.1-3
OneLoopTableInGroup 8.4-3 OneLoopTableInGroup 8.4-3
OneLoopWithMltGroup 8.4-6 OneLoopWithMltGroup 8.4-6
@ -342,8 +344,8 @@
OppositeLoop 4.12-1 OppositeLoop 4.12-1
OppositeQuasigroup 4.12-1 OppositeQuasigroup 4.12-1
Osborn loop 7.6-4 Osborn loop 7.6-4
Paige loop 9.8 Paige loop 9.9
PaigeLoop 9.8-1 PaigeLoop 9.9-1
Parent 6.1-1 Parent 6.1-1
PosInParent 6.1-3 PosInParent 6.1-3
Position 6.1-2 Position 6.1-2
@ -373,18 +375,20 @@
QuasigroupByRightFolder 4.7-1 QuasigroupByRightFolder 4.7-1
QuasigroupByRightSection 4.6-3 QuasigroupByRightSection 4.6-3
QuasigroupFromFile 4.5-1 QuasigroupFromFile 4.5-1
QuasigroupIsomorph 6.11-6
QuasigroupsUpToIsomorphism 6.11-3 QuasigroupsUpToIsomorphism 6.11-3
RandomLoop 4.9-1 RandomLoop 4.9-1
RandomNilpotentLoop 4.9-2 RandomNilpotentLoop 4.9-2
RandomQuasigroup 4.9-1 RandomQuasigroup 4.9-1
RC loop 7.4 RC loop 7.4
RCCLoop 9.6-1 RCCLoop 9.7-1
RelativeLeftMultiplicationGroup 6.4-2 RelativeLeftMultiplicationGroup 6.4-2
RelativeMultiplicationGroup 6.4-2 RelativeMultiplicationGroup 6.4-2
RelativeRightMultiplicationGroup 6.4-2 RelativeRightMultiplicationGroup 6.4-2
RightBolLoop 9.2-2 RightBolLoop 9.2-2
RightBolLoopByExactGroupFactorization 8.1-3 RightBolLoopByExactGroupFactorization 8.1-3
RightConjugacyClosedLoop 9.6-1 RightBruckLoop 9.3-2
RightConjugacyClosedLoop 9.7-1
RightCosets 6.2-6 RightCosets 6.2-6
RightDivision 5.2-1 RightDivision 5.2-1
RightDivision 5.2-1 RightDivision 5.2-1
@ -400,7 +404,7 @@
RightTransversal 6.2-7 RightTransversal 6.2-7
section, left 2.2 section, left 2.2
section, right 2.2 section, right 2.2
sedenion loop 9.11 sedenion loop 9.12
semisymmetric quasigroup 7.3-1 semisymmetric quasigroup 7.3-1
SetLoopElmName 3.4-1 SetLoopElmName 3.4-1
SetQuasigroupElmName 3.4-1 SetQuasigroupElmName 3.4-1
@ -408,12 +412,12 @@
simple loop 6.7-3 simple loop 6.7-3
Size 5.1-4 Size 5.1-4
SmallGeneratingSet 5.5-3 SmallGeneratingSet 5.5-3
SmallLoop 9.7-1 SmallLoop 9.8-1
solvability class 2.4 solvability class 2.4
solvable loop 2.4 solvable loop 2.4
Steiner loop 7.8-2 Steiner loop 7.8-2
Steiner quasigroup 7.3-4 Steiner quasigroup 7.3-4
SteinerLoop 9.5-1 SteinerLoop 9.6-1
strongly nilpotent loop 6.9-3 strongly nilpotent loop 6.9-3
subloop 2.3 subloop 2.3
Subloop 6.2-2 Subloop 6.2-2

54
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@ -37,7 +37,7 @@ alternative loop <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br />
alternative loop, left <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br /> alternative loop, left <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br />
alternative loop, right <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br /> alternative loop, right <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br />
antiautomorphic inverse property <a href="chap7_mj.html#X8538D4638232DB51">7.2-5</a><br /> antiautomorphic inverse property <a href="chap7_mj.html#X8538D4638232DB51">7.2-5</a><br />
<code class="func">AreEqualDiscriminators</code> <a href="chap6_mj.html#X812F0DEE7C896E18">6.11-9</a><br /> <code class="func">AreEqualDiscriminators</code> <a href="chap6_mj.html#X812F0DEE7C896E18">6.11-11</a><br />
<code class="func">AssociatedLeftBruckLoop</code> <a href="chap8_mj.html#X8664CA927DD73DBE">8.1-1</a><br /> <code class="func">AssociatedLeftBruckLoop</code> <a href="chap8_mj.html#X8664CA927DD73DBE">8.1-1</a><br />
<code class="func">AssociatedRightBruckLoop</code> <a href="chap8_mj.html#X8664CA927DD73DBE">8.1-1</a><br /> <code class="func">AssociatedRightBruckLoop</code> <a href="chap8_mj.html#X8664CA927DD73DBE">8.1-1</a><br />
associator <a href="chap2_mj.html#X7E0849977869E53D">2.5</a><br /> associator <a href="chap2_mj.html#X7E0849977869E53D">2.5</a><br />
@ -49,7 +49,7 @@ automorphic loop <a href="chap7_mj.html#X793B22EA8643C667">7.7</a><br />
automorphic loop, left <a href="chap7_mj.html#X793B22EA8643C667">7.7</a><br /> automorphic loop, left <a href="chap7_mj.html#X793B22EA8643C667">7.7</a><br />
automorphic loop, middle <a href="chap7_mj.html#X793B22EA8643C667">7.7</a><br /> automorphic loop, middle <a href="chap7_mj.html#X793B22EA8643C667">7.7</a><br />
automorphic loop, right <a href="chap7_mj.html#X793B22EA8643C667">7.7</a><br /> automorphic loop, right <a href="chap7_mj.html#X793B22EA8643C667">7.7</a><br />
<code class="func">AutomorphicLoop</code> <a href="chap9_mj.html#X784FFA9E7FDA9F43">9.10-1</a><br /> <code class="func">AutomorphicLoop</code> <a href="chap9_mj.html#X784FFA9E7FDA9F43">9.11-1</a><br />
<code class="func">AutomorphismGroup</code> <a href="chap6_mj.html#X87677B0787B4461A">6.11-5</a><br /> <code class="func">AutomorphismGroup</code> <a href="chap6_mj.html#X87677B0787B4461A">6.11-5</a><br />
Bol loop, left <a href="chap3_mj.html#X87E49ED884FA6DC4">3.3</a><br /> Bol loop, left <a href="chap3_mj.html#X87E49ED884FA6DC4">3.3</a><br />
Bol loop, left <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br /> Bol loop, left <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br />
@ -65,7 +65,7 @@ Cayley table <a href="chap4_mj.html#X7DE8405B82BC36A9">4.1</a><br />
Cayley table, canonical <a href="chap4_mj.html#X7971CCB87DAFF7B9">4.3-1</a><br /> Cayley table, canonical <a href="chap4_mj.html#X7971CCB87DAFF7B9">4.3-1</a><br />
<code class="func">CayleyTable</code> <a href="chap5_mj.html#X85457FA27DE7114D">5.1-2</a><br /> <code class="func">CayleyTable</code> <a href="chap5_mj.html#X85457FA27DE7114D">5.1-2</a><br />
<code class="func">CayleyTableByPerms</code> <a href="chap4_mj.html#X7F94C8DD7E1A3470">4.6-1</a><br /> <code class="func">CayleyTableByPerms</code> <a href="chap4_mj.html#X7F94C8DD7E1A3470">4.6-1</a><br />
<code class="func">CCLoop</code> <a href="chap9_mj.html#X798BC601843E8916">9.6-3</a><br /> <code class="func">CCLoop</code> <a href="chap9_mj.html#X798BC601843E8916">9.7-3</a><br />
center <a href="chap2_mj.html#X83EDF04F7952143F">2.3</a><br /> center <a href="chap2_mj.html#X83EDF04F7952143F">2.3</a><br />
<code class="func">Center</code> <a href="chap6_mj.html#X7C1FBE7A84DD4873">6.6-4</a><br /> <code class="func">Center</code> <a href="chap6_mj.html#X7C1FBE7A84DD4873">6.6-4</a><br />
central series, lower <a href="chap6_mj.html#X817BDBC2812992ED">6.9-5</a><br /> central series, lower <a href="chap6_mj.html#X817BDBC2812992ED">6.9-5</a><br />
@ -73,7 +73,7 @@ central series, upper <a href="chap2_mj.html#X869CBCE381E2C422">2.4</a><br />
Chein loop <a href="chap8_mj.html#X7CC6CDB786E9BBA0">8.2-3</a><br /> Chein loop <a href="chap8_mj.html#X7CC6CDB786E9BBA0">8.2-3</a><br />
cocycle <a href="chap4_mj.html#X8759431780AC81A9">4.8</a><br /> cocycle <a href="chap4_mj.html#X8759431780AC81A9">4.8</a><br />
code loop <a href="chap7_mj.html#X790FA1188087D5C1">7.8-1</a><br /> code loop <a href="chap7_mj.html#X790FA1188087D5C1">7.8-1</a><br />
<code class="func">CodeLoop</code> <a href="chap9_mj.html#X7DB4D3B27BB4D7EE">9.4-1</a><br /> <code class="func">CodeLoop</code> <a href="chap9_mj.html#X7DB4D3B27BB4D7EE">9.5-1</a><br />
commutant <a href="chap2_mj.html#X83EDF04F7952143F">2.3</a><br /> commutant <a href="chap2_mj.html#X83EDF04F7952143F">2.3</a><br />
<code class="func">Commutant</code> <a href="chap6_mj.html#X7C8428DE791F3CE1">6.6-3</a><br /> <code class="func">Commutant</code> <a href="chap6_mj.html#X7C8428DE791F3CE1">6.6-3</a><br />
commutator <a href="chap2_mj.html#X7E0849977869E53D">2.5</a><br /> commutator <a href="chap2_mj.html#X7E0849977869E53D">2.5</a><br />
@ -81,7 +81,7 @@ commutator <a href="chap2_mj.html#X7E0849977869E53D">2.5</a><br />
conjugacy closed loop <a href="chap7_mj.html#X8176B2C47A4629CD">7.6</a><br /> conjugacy closed loop <a href="chap7_mj.html#X8176B2C47A4629CD">7.6</a><br />
conjugacy closed loop, left <a href="chap7_mj.html#X8176B2C47A4629CD">7.6</a><br /> conjugacy closed loop, left <a href="chap7_mj.html#X8176B2C47A4629CD">7.6</a><br />
conjugacy closed loop, right <a href="chap7_mj.html#X8176B2C47A4629CD">7.6</a><br /> conjugacy closed loop, right <a href="chap7_mj.html#X8176B2C47A4629CD">7.6</a><br />
<code class="func">ConjugacyClosedLoop</code> <a href="chap9_mj.html#X798BC601843E8916">9.6-3</a><br /> <code class="func">ConjugacyClosedLoop</code> <a href="chap9_mj.html#X798BC601843E8916">9.7-3</a><br />
conjugation <a href="chap6_mj.html#X8740D61178ACD217">6.5</a><br /> conjugation <a href="chap6_mj.html#X8740D61178ACD217">6.5</a><br />
coset <a href="chap6_mj.html#X835F48248571364F">6.2-6</a><br /> coset <a href="chap6_mj.html#X835F48248571364F">6.2-6</a><br />
derived series <a href="chap2_mj.html#X869CBCE381E2C422">2.4</a><br /> derived series <a href="chap2_mj.html#X869CBCE381E2C422">2.4</a><br />
@ -90,7 +90,7 @@ derived subloop <a href="chap2_mj.html#X869CBCE381E2C422">2.4</a><br />
<code class="func">DerivedSubloop</code> <a href="chap6_mj.html#X7A82DC4680DAD67C">6.10-2</a><br /> <code class="func">DerivedSubloop</code> <a href="chap6_mj.html#X7A82DC4680DAD67C">6.10-2</a><br />
diassociative quasigroup <a href="chap7_mj.html#X872DCA027E1A4A1D">7.1-4</a><br /> diassociative quasigroup <a href="chap7_mj.html#X872DCA027E1A4A1D">7.1-4</a><br />
<code class="func">DirectProduct</code> <a href="chap4_mj.html#X861BA02C7902A4F4">4.11-1</a><br /> <code class="func">DirectProduct</code> <a href="chap4_mj.html#X861BA02C7902A4F4">4.11-1</a><br />
<code class="func">Discriminator</code> <a href="chap6_mj.html#X7D09D8957E4A0973">6.11-8</a><br /> <code class="func">Discriminator</code> <a href="chap6_mj.html#X7D09D8957E4A0973">6.11-10</a><br />
<code class="func">DisplayLibraryInfo</code> <a href="chap9_mj.html#X7A64372E81E713B4">9.1-3</a><br /> <code class="func">DisplayLibraryInfo</code> <a href="chap9_mj.html#X7A64372E81E713B4">9.1-3</a><br />
distributive quasigroup <a href="chap7_mj.html#X7B76FD6E878ED4F1">7.3-6</a><br /> distributive quasigroup <a href="chap7_mj.html#X7B76FD6E878ED4F1">7.3-6</a><br />
distributive quasigroup, left <a href="chap7_mj.html#X7B76FD6E878ED4F1">7.3-6</a><br /> distributive quasigroup, left <a href="chap7_mj.html#X7B76FD6E878ED4F1">7.3-6</a><br />
@ -137,7 +137,7 @@ inner mapping group, left <a href="chap2_mj.html#X7EC01B437CC2B2C9">2.2</a><br
inner mapping group, middle <a href="chap6_mj.html#X8740D61178ACD217">6.5</a><br /> inner mapping group, middle <a href="chap6_mj.html#X8740D61178ACD217">6.5</a><br />
inner mapping group, right <a href="chap2_mj.html#X7EC01B437CC2B2C9">2.2</a><br /> inner mapping group, right <a href="chap2_mj.html#X7EC01B437CC2B2C9">2.2</a><br />
<code class="func">InnerMappingGroup</code> <a href="chap6_mj.html#X82513A3B7C3A6420">6.5-3</a><br /> <code class="func">InnerMappingGroup</code> <a href="chap6_mj.html#X82513A3B7C3A6420">6.5-3</a><br />
<code class="func">InterestingLoop</code> <a href="chap9_mj.html#X87F24AD3811910D3">9.11-1</a><br /> <code class="func">InterestingLoop</code> <a href="chap9_mj.html#X87F24AD3811910D3">9.12-1</a><br />
<code class="func">IntoGroup</code> <a href="chap4_mj.html#X7B5C6C64831B866E">4.10-4</a><br /> <code class="func">IntoGroup</code> <a href="chap4_mj.html#X7B5C6C64831B866E">4.10-4</a><br />
<code class="func">IntoLoop</code> <a href="chap4_mj.html#X7A59C36683118E5A">4.10-3</a><br /> <code class="func">IntoLoop</code> <a href="chap4_mj.html#X7A59C36683118E5A">4.10-3</a><br />
<code class="func">IntoQuasigroup</code> <a href="chap4_mj.html#X84575A4B78CC545E">4.10-1</a><br /> <code class="func">IntoQuasigroup</code> <a href="chap4_mj.html#X84575A4B78CC545E">4.10-1</a><br />
@ -195,8 +195,8 @@ IsLoopElement <a href="chap3_mj.html#X86F02BBD87FEA1C6">3.1</a><br />
<code class="func">IsNilpotent</code> <a href="chap6_mj.html#X78A4B93781C96AAE">6.9-1</a><br /> <code class="func">IsNilpotent</code> <a href="chap6_mj.html#X78A4B93781C96AAE">6.9-1</a><br />
<code class="func">IsNormal</code> <a href="chap6_mj.html#X838186F9836F678C">6.7-1</a><br /> <code class="func">IsNormal</code> <a href="chap6_mj.html#X838186F9836F678C">6.7-1</a><br />
<code class="func">IsNuclearSquareLoop</code> <a href="chap7_mj.html#X796650088213229B">7.4-11</a><br /> <code class="func">IsNuclearSquareLoop</code> <a href="chap7_mj.html#X796650088213229B">7.4-11</a><br />
<code class="func">IsomorphicCopyByNormalSubloop</code> <a href="chap6_mj.html#X8121DE3A78795040">6.11-7</a><br /> <code class="func">IsomorphicCopyByNormalSubloop</code> <a href="chap6_mj.html#X8121DE3A78795040">6.11-9</a><br />
<code class="func">IsomorphicCopyByPerm</code> <a href="chap6_mj.html#X85B3E22679FD8D81">6.11-6</a><br /> <code class="func">IsomorphicCopyByPerm</code> <a href="chap6_mj.html#X85B3E22679FD8D81">6.11-8</a><br />
isomorphism <a href="chap2_mj.html#X791066ED7DD9F254">2.6</a><br /> isomorphism <a href="chap2_mj.html#X791066ED7DD9F254">2.6</a><br />
<code class="func">IsomorphismLoops</code> <a href="chap6_mj.html#X7D7B10D6836FCA9F">6.11-2</a><br /> <code class="func">IsomorphismLoops</code> <a href="chap6_mj.html#X7D7B10D6836FCA9F">6.11-2</a><br />
<code class="func">IsomorphismQuasigroups</code> <a href="chap6_mj.html#X801067F67E5292F7">6.11-1</a><br /> <code class="func">IsomorphismQuasigroups</code> <a href="chap6_mj.html#X801067F67E5292F7">6.11-1</a><br />
@ -232,16 +232,17 @@ IsQuasigroupElement <a href="chap3_mj.html#X86F02BBD87FEA1C6">3.1</a><br />
<code class="func">IsSubquasigroup</code> <a href="chap6_mj.html#X87AC8B7E80CE9260">6.2-3</a><br /> <code class="func">IsSubquasigroup</code> <a href="chap6_mj.html#X87AC8B7E80CE9260">6.2-3</a><br />
<code class="func">IsTotallySymmetric</code> <a href="chap7_mj.html#X834F809B8060B754">7.3-2</a><br /> <code class="func">IsTotallySymmetric</code> <a href="chap7_mj.html#X834F809B8060B754">7.3-2</a><br />
<code class="func">IsUnipotent</code> <a href="chap7_mj.html#X7CA3DCA07B6CB9BD">7.3-5</a><br /> <code class="func">IsUnipotent</code> <a href="chap7_mj.html#X7CA3DCA07B6CB9BD">7.3-5</a><br />
<code class="func">ItpSmallLoop</code> <a href="chap9_mj.html#X850C4C01817A098D">9.12-1</a><br /> <code class="func">ItpSmallLoop</code> <a href="chap9_mj.html#X850C4C01817A098D">9.13-1</a><br />
K loop, left <a href="chap7_mj.html#X85F1BD4280E44F5B">7.8-3</a><br /> K loop, left <a href="chap7_mj.html#X85F1BD4280E44F5B">7.8-3</a><br />
K loop, right <a href="chap7_mj.html#X857B373E7B4E0519">7.8-4</a><br /> K loop, right <a href="chap7_mj.html#X857B373E7B4E0519">7.8-4</a><br />
latin square <a href="chap2_mj.html#X80243DE5826583B8">2.1</a><br /> latin square <a href="chap2_mj.html#X80243DE5826583B8">2.1</a><br />
latin square <a href="chap4_mj.html#X7DE8405B82BC36A9">4.1</a><br /> latin square <a href="chap4_mj.html#X7DE8405B82BC36A9">4.1</a><br />
latin square, random <a href="chap4_mj.html#X7AE29A1A7AA5C25A">4.9</a><br /> latin square, random <a href="chap4_mj.html#X7AE29A1A7AA5C25A">4.9</a><br />
LC loop <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br /> LC loop <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br />
<code class="func">LCCLoop</code> <a href="chap9_mj.html#X80AB8B107D55FB19">9.6-2</a><br /> <code class="func">LCCLoop</code> <a href="chap9_mj.html#X80AB8B107D55FB19">9.7-2</a><br />
<code class="func">LeftBolLoop</code> <a href="chap9_mj.html#X7EE99F647C537994">9.2-1</a><br /> <code class="func">LeftBolLoop</code> <a href="chap9_mj.html#X7EE99F647C537994">9.2-1</a><br />
<code class="func">LeftConjugacyClosedLoop</code> <a href="chap9_mj.html#X80AB8B107D55FB19">9.6-2</a><br /> <code class="func">LeftBruckLoop</code> <a href="chap9_mj.html#X8290B01780F0FCD3">9.3-1</a><br />
<code class="func">LeftConjugacyClosedLoop</code> <a href="chap9_mj.html#X80AB8B107D55FB19">9.7-2</a><br />
<code class="func">LeftDivision</code> <a href="chap5_mj.html#X7D5956967BCC1834">5.2-1</a><br /> <code class="func">LeftDivision</code> <a href="chap5_mj.html#X7D5956967BCC1834">5.2-1</a><br />
<code class="func">LeftDivision</code> <a href="chap5_mj.html#X7D5956967BCC1834">5.2-1</a><br /> <code class="func">LeftDivision</code> <a href="chap5_mj.html#X7D5956967BCC1834">5.2-1</a><br />
<code class="func">LeftDivision</code> <a href="chap5_mj.html#X7D5956967BCC1834">5.2-1</a><br /> <code class="func">LeftDivision</code> <a href="chap5_mj.html#X7D5956967BCC1834">5.2-1</a><br />
@ -260,7 +261,7 @@ loop, Chein <a href="chap8_mj.html#X7CC6CDB786E9BBA0">8.2-3</a><br />
loop, LC <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br /> loop, LC <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br />
loop, Moufang <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br /> loop, Moufang <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br />
loop, Osborn <a href="chap7_mj.html#X8655956878205FC1">7.6-4</a><br /> loop, Osborn <a href="chap7_mj.html#X8655956878205FC1">7.6-4</a><br />
loop, Paige <a href="chap9_mj.html#X8135C8FD8714C606">9.8</a><br /> loop, Paige <a href="chap9_mj.html#X8135C8FD8714C606">9.9</a><br />
loop, RC <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br /> loop, RC <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br />
loop, Steiner <a href="chap7_mj.html#X793600C9801F4F62">7.8-2</a><br /> loop, Steiner <a href="chap7_mj.html#X793600C9801F4F62">7.8-2</a><br />
loop, alternative <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br /> loop, alternative <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br />
@ -285,7 +286,7 @@ loop, middle nuclear square <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><b
loop, nilpotent <a href="chap2_mj.html#X869CBCE381E2C422">2.4</a><br /> loop, nilpotent <a href="chap2_mj.html#X869CBCE381E2C422">2.4</a><br />
loop, nilpotent <a href="chap4_mj.html#X817132C887D3FD3A">4.9-2</a><br /> loop, nilpotent <a href="chap4_mj.html#X817132C887D3FD3A">4.9-2</a><br />
loop, nuclear square <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br /> loop, nuclear square <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br />
loop, octonion <a href="chap9_mj.html#X81E82098822543EE">9.3-1</a><br /> loop, octonion <a href="chap9_mj.html#X81E82098822543EE">9.4-1</a><br />
loop, of Bol-Moufang type <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br /> loop, of Bol-Moufang type <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br />
loop, power alternative <a href="chap7_mj.html#X83A501387E1AC371">7.5</a><br /> loop, power alternative <a href="chap7_mj.html#X83A501387E1AC371">7.5</a><br />
loop, power associative <a href="chap5_mj.html#X7D44470C7DA59C1C">5.1-5</a><br /> loop, power associative <a href="chap5_mj.html#X7D44470C7DA59C1C">5.1-5</a><br />
@ -297,7 +298,7 @@ loop, right automorphic <a href="chap7_mj.html#X793B22EA8643C667">7.7</a><br />
loop, right conjugacy closed <a href="chap7_mj.html#X8176B2C47A4629CD">7.6</a><br /> loop, right conjugacy closed <a href="chap7_mj.html#X8176B2C47A4629CD">7.6</a><br />
loop, right nuclear square <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br /> loop, right nuclear square <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br />
loop, right power alternative <a href="chap7_mj.html#X83A501387E1AC371">7.5</a><br /> loop, right power alternative <a href="chap7_mj.html#X83A501387E1AC371">7.5</a><br />
loop, sedenion <a href="chap9_mj.html#X843BD73F788049F7">9.11</a><br /> loop, sedenion <a href="chap9_mj.html#X843BD73F788049F7">9.12</a><br />
loop, simple <a href="chap3_mj.html#X87E49ED884FA6DC4">3.3</a><br /> loop, simple <a href="chap3_mj.html#X87E49ED884FA6DC4">3.3</a><br />
loop, simple <a href="chap6_mj.html#X7D8E63A7824037CC">6.7-3</a><br /> loop, simple <a href="chap6_mj.html#X7D8E63A7824037CC">6.7-3</a><br />
loop, solvable <a href="chap2_mj.html#X869CBCE381E2C422">2.4</a><br /> loop, solvable <a href="chap2_mj.html#X869CBCE381E2C422">2.4</a><br />
@ -312,6 +313,7 @@ loop table <a href="chap4_mj.html#X7DE8405B82BC36A9">4.1</a><br />
<code class="func">LoopByRightFolder</code> <a href="chap4_mj.html#X83168E62861F70AB">4.7-1</a><br /> <code class="func">LoopByRightFolder</code> <a href="chap4_mj.html#X83168E62861F70AB">4.7-1</a><br />
<code class="func">LoopByRightSection</code> <a href="chap4_mj.html#X80B436ED7CC0749E">4.6-3</a><br /> <code class="func">LoopByRightSection</code> <a href="chap4_mj.html#X80B436ED7CC0749E">4.6-3</a><br />
<code class="func">LoopFromFile</code> <a href="chap4_mj.html#X81A1DB918057933E">4.5-1</a><br /> <code class="func">LoopFromFile</code> <a href="chap4_mj.html#X81A1DB918057933E">4.5-1</a><br />
<code class="func">LoopIsomorph</code> <a href="chap6_mj.html#X7BD1AC32851286EA">6.11-7</a><br />
<code class="func">LoopMG2</code> <a href="chap8_mj.html#X7CC6CDB786E9BBA0">8.2-3</a><br /> <code class="func">LoopMG2</code> <a href="chap8_mj.html#X7CC6CDB786E9BBA0">8.2-3</a><br />
<code class="func">LoopsUpToIsomorphism</code> <a href="chap6_mj.html#X8308F38283C61B20">6.11-4</a><br /> <code class="func">LoopsUpToIsomorphism</code> <a href="chap6_mj.html#X8308F38283C61B20">6.11-4</a><br />
<code class="func">LoopsUpToIsotopism</code> <a href="chap6_mj.html#X841E540B7A7EF29F">6.12-2</a><br /> <code class="func">LoopsUpToIsotopism</code> <a href="chap6_mj.html#X841E540B7A7EF29F">6.12-2</a><br />
@ -325,7 +327,7 @@ modification, Moufang <a href="chap8_mj.html#X819F82737C2A860D">8.2</a><br />
modification, cyclic <a href="chap8_mj.html#X7B3165C083709831">8.2-1</a><br /> modification, cyclic <a href="chap8_mj.html#X7B3165C083709831">8.2-1</a><br />
modification, dihedral <a href="chap8_mj.html#X7D7717C587BC2D1E">8.2-2</a><br /> modification, dihedral <a href="chap8_mj.html#X7D7717C587BC2D1E">8.2-2</a><br />
Moufang loop <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br /> Moufang loop <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br />
<code class="func">MoufangLoop</code> <a href="chap9_mj.html#X81E82098822543EE">9.3-1</a><br /> <code class="func">MoufangLoop</code> <a href="chap9_mj.html#X81E82098822543EE">9.4-1</a><br />
multiplication group <a href="chap2_mj.html#X7EC01B437CC2B2C9">2.2</a><br /> multiplication group <a href="chap2_mj.html#X7EC01B437CC2B2C9">2.2</a><br />
multiplication group, left <a href="chap2_mj.html#X7EC01B437CC2B2C9">2.2</a><br /> multiplication group, left <a href="chap2_mj.html#X7EC01B437CC2B2C9">2.2</a><br />
multiplication group, relative <a href="chap6_mj.html#X847256B779E1E7E5">6.4-2</a><br /> multiplication group, relative <a href="chap6_mj.html#X847256B779E1E7E5">6.4-2</a><br />
@ -341,7 +343,7 @@ nilpotence class <a href="chap2_mj.html#X869CBCE381E2C422">2.4</a><br />
<code class="func">NilpotencyClassOfLoop</code> <a href="chap6_mj.html#X7D5FC62581A99482">6.9-2</a><br /> <code class="func">NilpotencyClassOfLoop</code> <a href="chap6_mj.html#X7D5FC62581A99482">6.9-2</a><br />
nilpotent loop <a href="chap2_mj.html#X869CBCE381E2C422">2.4</a><br /> nilpotent loop <a href="chap2_mj.html#X869CBCE381E2C422">2.4</a><br />
nilpotent loop, strongly <a href="chap6_mj.html#X7E7C2D117B55F6A0">6.9-3</a><br /> nilpotent loop, strongly <a href="chap6_mj.html#X7E7C2D117B55F6A0">6.9-3</a><br />
<code class="func">NilpotentLoop</code> <a href="chap9_mj.html#X7A9C960D86E2AD28">9.9-1</a><br /> <code class="func">NilpotentLoop</code> <a href="chap9_mj.html#X7A9C960D86E2AD28">9.10-1</a><br />
normal closure <a href="chap6_mj.html#X7BDEA0A98720D1BB">6.7-2</a><br /> normal closure <a href="chap6_mj.html#X7BDEA0A98720D1BB">6.7-2</a><br />
normal subloop <a href="chap6_mj.html#X838186F9836F678C">6.7-1</a><br /> normal subloop <a href="chap6_mj.html#X838186F9836F678C">6.7-1</a><br />
<code class="func">NormalClosure</code> <a href="chap6_mj.html#X7BDEA0A98720D1BB">6.7-2</a><br /> <code class="func">NormalClosure</code> <a href="chap6_mj.html#X7BDEA0A98720D1BB">6.7-2</a><br />
@ -358,7 +360,7 @@ nucleus, middle <a href="chap2_mj.html#X83EDF04F7952143F">2.3</a><br />
nucleus, right <a href="chap2_mj.html#X83EDF04F7952143F">2.3</a><br /> nucleus, right <a href="chap2_mj.html#X83EDF04F7952143F">2.3</a><br />
<code class="func">NucleusOfLoop</code> <a href="chap6_mj.html#X84D389677A91C290">6.6-2</a><br /> <code class="func">NucleusOfLoop</code> <a href="chap6_mj.html#X84D389677A91C290">6.6-2</a><br />
<code class="func">NucleusOfQuasigroup</code> <a href="chap6_mj.html#X84D389677A91C290">6.6-2</a><br /> <code class="func">NucleusOfQuasigroup</code> <a href="chap6_mj.html#X84D389677A91C290">6.6-2</a><br />
octonion loop <a href="chap9_mj.html#X81E82098822543EE">9.3-1</a><br /> octonion loop <a href="chap9_mj.html#X81E82098822543EE">9.4-1</a><br />
<code class="func">One</code> <a href="chap5_mj.html#X8129A6877FFD804B">5.1-3</a><br /> <code class="func">One</code> <a href="chap5_mj.html#X8129A6877FFD804B">5.1-3</a><br />
<code class="func">OneLoopTableInGroup</code> <a href="chap8_mj.html#X7BFFC66A824BA6AA">8.4-3</a><br /> <code class="func">OneLoopTableInGroup</code> <a href="chap8_mj.html#X7BFFC66A824BA6AA">8.4-3</a><br />
<code class="func">OneLoopWithMltGroup</code> <a href="chap8_mj.html#X8266DE05824226E6">8.4-6</a><br /> <code class="func">OneLoopWithMltGroup</code> <a href="chap8_mj.html#X8266DE05824226E6">8.4-6</a><br />
@ -368,8 +370,8 @@ opposite quasigroup <a href="chap4_mj.html#X7865FC8D7854C2E3">4.12</a><br />
<code class="func">OppositeLoop</code> <a href="chap4_mj.html#X87B6AED47EE2BCD3">4.12-1</a><br /> <code class="func">OppositeLoop</code> <a href="chap4_mj.html#X87B6AED47EE2BCD3">4.12-1</a><br />
<code class="func">OppositeQuasigroup</code> <a href="chap4_mj.html#X87B6AED47EE2BCD3">4.12-1</a><br /> <code class="func">OppositeQuasigroup</code> <a href="chap4_mj.html#X87B6AED47EE2BCD3">4.12-1</a><br />
Osborn loop <a href="chap7_mj.html#X8655956878205FC1">7.6-4</a><br /> Osborn loop <a href="chap7_mj.html#X8655956878205FC1">7.6-4</a><br />
Paige loop <a href="chap9_mj.html#X8135C8FD8714C606">9.8</a><br /> Paige loop <a href="chap9_mj.html#X8135C8FD8714C606">9.9</a><br />
<code class="func">PaigeLoop</code> <a href="chap9_mj.html#X7FCF4D6B7AD66D74">9.8-1</a><br /> <code class="func">PaigeLoop</code> <a href="chap9_mj.html#X7FCF4D6B7AD66D74">9.9-1</a><br />
<code class="func">Parent</code> <a href="chap6_mj.html#X7BC856CC7F116BB0">6.1-1</a><br /> <code class="func">Parent</code> <a href="chap6_mj.html#X7BC856CC7F116BB0">6.1-1</a><br />
<code class="func">PosInParent</code> <a href="chap6_mj.html#X832295DE866E44EE">6.1-3</a><br /> <code class="func">PosInParent</code> <a href="chap6_mj.html#X832295DE866E44EE">6.1-3</a><br />
<code class="func">Position</code> <a href="chap6_mj.html#X79975EC6783B4293">6.1-2</a><br /> <code class="func">Position</code> <a href="chap6_mj.html#X79975EC6783B4293">6.1-2</a><br />
@ -399,18 +401,20 @@ quasigroup table <a href="chap4_mj.html#X7DE8405B82BC36A9">4.1</a><br />
<code class="func">QuasigroupByRightFolder</code> <a href="chap4_mj.html#X83168E62861F70AB">4.7-1</a><br /> <code class="func">QuasigroupByRightFolder</code> <a href="chap4_mj.html#X83168E62861F70AB">4.7-1</a><br />
<code class="func">QuasigroupByRightSection</code> <a href="chap4_mj.html#X80B436ED7CC0749E">4.6-3</a><br /> <code class="func">QuasigroupByRightSection</code> <a href="chap4_mj.html#X80B436ED7CC0749E">4.6-3</a><br />
<code class="func">QuasigroupFromFile</code> <a href="chap4_mj.html#X81A1DB918057933E">4.5-1</a><br /> <code class="func">QuasigroupFromFile</code> <a href="chap4_mj.html#X81A1DB918057933E">4.5-1</a><br />
<code class="func">QuasigroupIsomorph</code> <a href="chap6_mj.html#X7A42812B7B027DD4">6.11-6</a><br />
<code class="func">QuasigroupsUpToIsomorphism</code> <a href="chap6_mj.html#X82373C5479574F22">6.11-3</a><br /> <code class="func">QuasigroupsUpToIsomorphism</code> <a href="chap6_mj.html#X82373C5479574F22">6.11-3</a><br />
<code class="func">RandomLoop</code> <a href="chap4_mj.html#X8271C0F5786B6FA9">4.9-1</a><br /> <code class="func">RandomLoop</code> <a href="chap4_mj.html#X8271C0F5786B6FA9">4.9-1</a><br />
<code class="func">RandomNilpotentLoop</code> <a href="chap4_mj.html#X817132C887D3FD3A">4.9-2</a><br /> <code class="func">RandomNilpotentLoop</code> <a href="chap4_mj.html#X817132C887D3FD3A">4.9-2</a><br />
<code class="func">RandomQuasigroup</code> <a href="chap4_mj.html#X8271C0F5786B6FA9">4.9-1</a><br /> <code class="func">RandomQuasigroup</code> <a href="chap4_mj.html#X8271C0F5786B6FA9">4.9-1</a><br />
RC loop <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br /> RC loop <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br />
<code class="func">RCCLoop</code> <a href="chap9_mj.html#X806B2DE67990E42F">9.6-1</a><br /> <code class="func">RCCLoop</code> <a href="chap9_mj.html#X806B2DE67990E42F">9.7-1</a><br />
<code class="func">RelativeLeftMultiplicationGroup</code> <a href="chap6_mj.html#X847256B779E1E7E5">6.4-2</a><br /> <code class="func">RelativeLeftMultiplicationGroup</code> <a href="chap6_mj.html#X847256B779E1E7E5">6.4-2</a><br />
<code class="func">RelativeMultiplicationGroup</code> <a href="chap6_mj.html#X847256B779E1E7E5">6.4-2</a><br /> <code class="func">RelativeMultiplicationGroup</code> <a href="chap6_mj.html#X847256B779E1E7E5">6.4-2</a><br />
<code class="func">RelativeRightMultiplicationGroup</code> <a href="chap6_mj.html#X847256B779E1E7E5">6.4-2</a><br /> <code class="func">RelativeRightMultiplicationGroup</code> <a href="chap6_mj.html#X847256B779E1E7E5">6.4-2</a><br />
<code class="func">RightBolLoop</code> <a href="chap9_mj.html#X8774304282654C58">9.2-2</a><br /> <code class="func">RightBolLoop</code> <a href="chap9_mj.html#X8774304282654C58">9.2-2</a><br />
<code class="func">RightBolLoopByExactGroupFactorization</code> <a href="chap8_mj.html#X7DCA64807F899127">8.1-3</a><br /> <code class="func">RightBolLoopByExactGroupFactorization</code> <a href="chap8_mj.html#X7DCA64807F899127">8.1-3</a><br />
<code class="func">RightConjugacyClosedLoop</code> <a href="chap9_mj.html#X806B2DE67990E42F">9.6-1</a><br /> <code class="func">RightBruckLoop</code> <a href="chap9_mj.html#X798DD7CF871F648F">9.3-2</a><br />
<code class="func">RightConjugacyClosedLoop</code> <a href="chap9_mj.html#X806B2DE67990E42F">9.7-1</a><br />
<code class="func">RightCosets</code> <a href="chap6_mj.html#X835F48248571364F">6.2-6</a><br /> <code class="func">RightCosets</code> <a href="chap6_mj.html#X835F48248571364F">6.2-6</a><br />
<code class="func">RightDivision</code> <a href="chap5_mj.html#X7D5956967BCC1834">5.2-1</a><br /> <code class="func">RightDivision</code> <a href="chap5_mj.html#X7D5956967BCC1834">5.2-1</a><br />
<code class="func">RightDivision</code> <a href="chap5_mj.html#X7D5956967BCC1834">5.2-1</a><br /> <code class="func">RightDivision</code> <a href="chap5_mj.html#X7D5956967BCC1834">5.2-1</a><br />
@ -426,7 +430,7 @@ RC loop <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br />
<code class="func">RightTransversal</code> <a href="chap6_mj.html#X85C65D06822E716F">6.2-7</a><br /> <code class="func">RightTransversal</code> <a href="chap6_mj.html#X85C65D06822E716F">6.2-7</a><br />
section, left <a href="chap2_mj.html#X7EC01B437CC2B2C9">2.2</a><br /> section, left <a href="chap2_mj.html#X7EC01B437CC2B2C9">2.2</a><br />
section, right <a href="chap2_mj.html#X7EC01B437CC2B2C9">2.2</a><br /> section, right <a href="chap2_mj.html#X7EC01B437CC2B2C9">2.2</a><br />
sedenion loop <a href="chap9_mj.html#X843BD73F788049F7">9.11</a><br /> sedenion loop <a href="chap9_mj.html#X843BD73F788049F7">9.12</a><br />
semisymmetric quasigroup <a href="chap7_mj.html#X834848ED85F9012B">7.3-1</a><br /> semisymmetric quasigroup <a href="chap7_mj.html#X834848ED85F9012B">7.3-1</a><br />
<code class="func">SetLoopElmName</code> <a href="chap3_mj.html#X7A7EB1B579273D07">3.4-1</a><br /> <code class="func">SetLoopElmName</code> <a href="chap3_mj.html#X7A7EB1B579273D07">3.4-1</a><br />
<code class="func">SetQuasigroupElmName</code> <a href="chap3_mj.html#X7A7EB1B579273D07">3.4-1</a><br /> <code class="func">SetQuasigroupElmName</code> <a href="chap3_mj.html#X7A7EB1B579273D07">3.4-1</a><br />
@ -434,12 +438,12 @@ simple loop <a href="chap3_mj.html#X87E49ED884FA6DC4">3.3</a><br />
simple loop <a href="chap6_mj.html#X7D8E63A7824037CC">6.7-3</a><br /> simple loop <a href="chap6_mj.html#X7D8E63A7824037CC">6.7-3</a><br />
<code class="func">Size</code> <a href="chap5_mj.html#X858ADA3B7A684421">5.1-4</a><br /> <code class="func">Size</code> <a href="chap5_mj.html#X858ADA3B7A684421">5.1-4</a><br />
<code class="func">SmallGeneratingSet</code> <a href="chap5_mj.html#X814DBABC878D5232">5.5-3</a><br /> <code class="func">SmallGeneratingSet</code> <a href="chap5_mj.html#X814DBABC878D5232">5.5-3</a><br />
<code class="func">SmallLoop</code> <a href="chap9_mj.html#X7C6EE23E84CD87D3">9.7-1</a><br /> <code class="func">SmallLoop</code> <a href="chap9_mj.html#X7C6EE23E84CD87D3">9.8-1</a><br />
solvability class <a href="chap2_mj.html#X869CBCE381E2C422">2.4</a><br /> solvability class <a href="chap2_mj.html#X869CBCE381E2C422">2.4</a><br />
solvable loop <a href="chap2_mj.html#X869CBCE381E2C422">2.4</a><br /> solvable loop <a href="chap2_mj.html#X869CBCE381E2C422">2.4</a><br />
Steiner loop <a href="chap7_mj.html#X793600C9801F4F62">7.8-2</a><br /> Steiner loop <a href="chap7_mj.html#X793600C9801F4F62">7.8-2</a><br />
Steiner quasigroup <a href="chap7_mj.html#X83DE7DD77C056C1F">7.3-4</a><br /> Steiner quasigroup <a href="chap7_mj.html#X83DE7DD77C056C1F">7.3-4</a><br />
<code class="func">SteinerLoop</code> <a href="chap9_mj.html#X87C235457E859AF4">9.5-1</a><br /> <code class="func">SteinerLoop</code> <a href="chap9_mj.html#X87C235457E859AF4">9.6-1</a><br />
strongly nilpotent loop <a href="chap6_mj.html#X7E7C2D117B55F6A0">6.9-3</a><br /> strongly nilpotent loop <a href="chap6_mj.html#X7E7C2D117B55F6A0">6.9-3</a><br />
subloop <a href="chap2_mj.html#X83EDF04F7952143F">2.3</a><br /> subloop <a href="chap2_mj.html#X83EDF04F7952143F">2.3</a><br />
<code class="func">Subloop</code> <a href="chap6_mj.html#X84E6744E804AE830">6.2-2</a><br /> <code class="func">Subloop</code> <a href="chap6_mj.html#X84E6744E804AE830">6.2-2</a><br />

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@ -1,161 +0,0 @@
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\newblock Moufang loops that share associator and three quarters of their
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E.~G. Goodaire, S.~May, and M.~Raman.
\newblock {\em The {M}oufang loops of order less than 64}.
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\newblock {\em LMS J. Comput. Math.}, 14:200{\textendash}213, 2011.
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P.~Jedli{\v c}ka, M.~Kinyon, and P.~Vojt{\v e}chovsk{\a'y}.
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@ -260,6 +260,27 @@ Publishers, 1999.
MRNUMBER = {1689624 (2000a:20147)}, MRNUMBER = {1689624 (2000a:20147)},
} }
\bibitem{Greer}
Mark Greer.
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Comm. Algebra {42} (2014), 3682--3697.
@article {Greer,
AUTHOR = {Greer, Mark},
TITLE = {A class of loops categorically isomorphic to {B}ruck loops of odd order},
JOURNAL = {Comm. Algebra},
FJOURNAL = {Communications in Algebra},
VOLUME = {42},
YEAR = {2014},
NUMBER = {8},
PAGES = {3682--3697},
ISSN = {0092-7872},
MRCLASS = {20N05},
MRNUMBER = {3196069},
MRREVIEWER = {Anil Kumar V.},
URL = {https://doi.org/10.1080/00927872.2013.791304},
}
\bibitem{GrKiNa} \bibitem{GrKiNa}
Alexander Grishkov, Michael Kinyon and G\'abor Nagy. Alexander Grishkov, Michael Kinyon and G\'abor Nagy.
\newblock {\it Solvability of commutative automorphic loops}, \newblock {\it Solvability of commutative automorphic loops},
@ -591,6 +612,19 @@ preprint.
PAGES = {423--428}, PAGES = {423--428},
} }
\bibitem{StuhlVojtechovsky}
Izabella Stuhl and Petr Vojt\v{e}chovsk\'y.
\newblock {\it Involutory latin quandles, Bruck loops and commutative automorphic loops of odd prime power order},
preprint, 2017.
@article {StuhlVojtechovsky,
AUTHOR = {Stuhl, Izabella and Vojt{\v{e}}chovsk{\'y}, Petr},
TITLE = {Involutory latin quandles, Bruck loops and commutative automorphic loops of odd prime power order},
JOURNAL = {},
YEAR = {2017},
NOTE = {preprint},
}
\bibitem{Vo} \bibitem{Vo}
Petr Vojt\v{e}chovsk\'y. Petr Vojt\v{e}chovsk\'y.
\newblock {\it Toward the classification of Moufang loops of order $64$}, \newblock {\it Toward the classification of Moufang loops of order $64$},

5
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@ -1,5 +0,0 @@
This is BibTeX, Version 0.99dThe top-level auxiliary file: loops.aux
The style file: alpha.bst
Database file #1: loops_bib.xml.bib
Warning--empty journal in KiNaVo2015
(There was 1 warning)

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@ -1,35 +0,0 @@
\backcite {Br}{{8}{2}{chapter.2}}
\backcite {Pf}{{8}{2}{chapter.2}}
\backcite {JaMa}{{19}{4.9}{section.4.9}}
\backcite {Vo}{{35}{6.11.8}{subsection.6.11.8}}
\backcite {Ar}{{37}{7.2.4}{subsection.7.2.4}}
\backcite {Fe}{{39}{7.4}{section.7.4}}
\backcite {PhiVoj}{{39}{7.4}{section.7.4}}
\backcite {PhiVoj}{{39}{7.4}{section.7.4}}
\backcite {BrPa}{{43}{7.7}{section.7.7}}
\backcite {BrPa}{{43}{7.7}{section.7.7}}
\backcite {JoKiNaVo}{{43}{7.7}{section.7.7}}
\backcite {KiKuPh}{{43}{7.7}{section.7.7}}
\backcite {KiKuPhVo}{{43}{7.7}{section.7.7}}
\backcite {GrKiNa}{{43}{7.7}{section.7.7}}
\backcite {VoQRS}{{43}{7.7}{section.7.7}}
\backcite {JeKiVo}{{43}{7.7}{section.7.7}}
\backcite {BaGrVo}{{43}{7.7}{section.7.7}}
\backcite {VoQRS}{{43}{7.7}{section.7.7}}
\backcite {DrapalCD}{{46}{8.2}{section.8.2}}
\backcite {DrVo}{{46}{8.2}{section.8.2}}
\backcite {NaVo2003}{{46}{8.3}{section.8.3}}
\backcite {Mo}{{50}{9.2}{section.9.2}}
\backcite {KiNaVo2015}{{50}{9.2}{section.9.2}}
\backcite {Go}{{50}{9.3.1}{subsection.9.3.1}}
\backcite {NaVo2007}{{50}{9.3.1}{subsection.9.3.1}}
\backcite {SlZe2011}{{50}{9.3.1}{subsection.9.3.1}}
\backcite {NaVo2007}{{51}{9.4}{section.9.4}}
\backcite {CoRo}{{51}{9.5}{section.9.5}}
\backcite {Artic}{{51}{9.6}{section.9.6}}
\backcite {Kun}{{52}{9.6.2}{subsection.9.6.2}}
\backcite {CsDr}{{52}{9.6.2}{subsection.9.6.2}}
\backcite {Wi}{{52}{9.6.2}{subsection.9.6.2}}
\backcite {Kun}{{52}{9.6.2}{subsection.9.6.2}}
\backcite {Li}{{53}{9.8}{section.9.8}}
\backcite {DaVo}{{53}{9.9}{section.9.9}}

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@ -1,427 +0,0 @@
\indexentry{groupoid|hyperpage}{8}
\indexentry{magma|hyperpage}{8}
\indexentry{neutral element|hyperpage}{8}
\indexentry{identity!element|hyperpage}{8}
\indexentry{inverse!two-sided|hyperpage}{8}
\indexentry{group|hyperpage}{8}
\indexentry{quasigroup|hyperpage}{8}
\indexentry{latin square|hyperpage}{8}
\indexentry{loop|hyperpage}{8}
\indexentry{translation!left|hyperpage}{8}
\indexentry{translation!right|hyperpage}{8}
\indexentry{division!left|hyperpage}{8}
\indexentry{division!right|hyperpage}{8}
\indexentry{section!left|hyperpage}{8}
\indexentry{section!right|hyperpage}{8}
\indexentry{multiplication group!left|hyperpage}{9}
\indexentry{multiplication group!right|hyperpage}{9}
\indexentry{multiplication group|hyperpage}{9}
\indexentry{inner mapping group!left|hyperpage}{9}
\indexentry{inner mapping group!right|hyperpage}{9}
\indexentry{inner mapping group|hyperpage}{9}
\indexentry{subquasigroup|hyperpage}{9}
\indexentry{subloop|hyperpage}{9}
\indexentry{nucleus!left|hyperpage}{9}
\indexentry{nucleus!middle|hyperpage}{9}
\indexentry{nucleus!right|hyperpage}{9}
\indexentry{nucleus|hyperpage}{9}
\indexentry{commutant|hyperpage}{9}
\indexentry{center|hyperpage}{9}
\indexentry{subloop!normal|hyperpage}{9}
\indexentry{nilpotence class|hyperpage}{9}
\indexentry{nilpotent loop|hyperpage}{9}
\indexentry{loop!nilpotent|hyperpage}{9}
\indexentry{central series!upper|hyperpage}{9}
\indexentry{derived subloop|hyperpage}{9}
\indexentry{solvability class|hyperpage}{9}
\indexentry{solvable loop|hyperpage}{9}
\indexentry{loop!solvable|hyperpage}{9}
\indexentry{derived series|hyperpage}{9}
\indexentry{commutator|hyperpage}{9}
\indexentry{associator|hyperpage}{9}
\indexentry{associator subloop|hyperpage}{9}
\indexentry{homomorphism|hyperpage}{9}
\indexentry{isomorphism|hyperpage}{9}
\indexentry{homotopism|hyperpage}{10}
\indexentry{isotopism|hyperpage}{10}
\indexentry{isotopism!principal|hyperpage}{10}
\indexentry{loop isotope!principal|hyperpage}{10}
\indexentry{IsQuasigroupElement|hyperpage}{11}
\indexentry{IsLoopElement|hyperpage}{11}
\indexentry{IsQuasigroup|hyperpage}{11}
\indexentry{IsLoop|hyperpage}{11}
\indexentry{Bol loop!left|hyperpage}{12}
\indexentry{loop!left Bol|hyperpage}{12}
\indexentry{simple loop|hyperpage}{12}
\indexentry{loop!simple|hyperpage}{12}
\indexentry{SetQuasigroupElmName@\texttt {SetQuasigroupElmName}|hyperpage}{13}
\indexentry{SetLoopElmName@\texttt {SetLoopElmName}|hyperpage}{13}
\indexentry{Cayley table|hyperpage}{14}
\indexentry{multiplication table|hyperpage}{14}
\indexentry{quasigroup table|hyperpage}{14}
\indexentry{latin square|hyperpage}{14}
\indexentry{loop table|hyperpage}{14}
\indexentry{IsQuasigroupTable@\texttt {IsQuasigroupTable}|hyperpage}{14}
\indexentry{IsQuasigroupCayleyTable@\texttt {IsQuasigroupCayleyTable}|hyperpage}{14}
\indexentry{IsLoopTable@\texttt {IsLoopTable}|hyperpage}{14}
\indexentry{IsLoopCayleyTable@\texttt {IsLoopCayleyTable}|hyperpage}{14}
\indexentry{CanonicalCayleyTable@\texttt {CanonicalCayleyTable}|hyperpage}{15}
\indexentry{Cayley table!canonical|hyperpage}{15}
\indexentry{CanonicalCopy@\texttt {CanonicalCopy}|hyperpage}{15}
\indexentry{NormalizedQuasigroupTable@\texttt {NormalizedQuasigroupTable}|hyperpage}{15}
\indexentry{QuasigroupByCayleyTable@\texttt {QuasigroupByCayleyTable}|hyperpage}{15}
\indexentry{LoopByCayleyTable@\texttt {LoopByCayleyTable}|hyperpage}{15}
\indexentry{QuasigroupFromFile@\texttt {QuasigroupFromFile}|hyperpage}{17}
\indexentry{LoopFromFile@\texttt {LoopFromFile}|hyperpage}{17}
\indexentry{CayleyTableByPerms@\texttt {CayleyTableByPerms}|hyperpage}{17}
\indexentry{QuasigroupByLeftSection@\texttt {QuasigroupByLeftSection}|hyperpage}{17}
\indexentry{LoopByLeftSection@\texttt {LoopByLeftSection}|hyperpage}{17}
\indexentry{QuasigroupByRightSection@\texttt {QuasigroupByRightSection}|hyperpage}{17}
\indexentry{LoopByRightSection@\texttt {LoopByRightSection}|hyperpage}{17}
\indexentry{folder!quasigroup|hyperpage}{18}
\indexentry{QuasigroupByRightFolder@\texttt {QuasigroupByRightFolder}|hyperpage}{18}
\indexentry{LoopByRightFolder@\texttt {LoopByRightFolder}|hyperpage}{18}
\indexentry{extension|hyperpage}{18}
\indexentry{extension!nuclear|hyperpage}{18}
\indexentry{cocycle|hyperpage}{18}
\indexentry{NuclearExtension@\texttt {NuclearExtension}|hyperpage}{18}
\indexentry{LoopByExtension@\texttt {LoopByExtension}|hyperpage}{18}
\indexentry{latin square!random|hyperpage}{19}
\indexentry{RandomQuasigroup@\texttt {RandomQuasigroup}|hyperpage}{19}
\indexentry{RandomLoop@\texttt {RandomLoop}|hyperpage}{19}
\indexentry{RandomNilpotentLoop@\texttt {RandomNilpotentLoop}|hyperpage}{19}
\indexentry{loop!nilpotent|hyperpage}{19}
\indexentry{IntoQuasigroup@\texttt {IntoQuasigroup}|hyperpage}{20}
\indexentry{PrincipalLoopIsotope@\texttt {PrincipalLoopIsotope}|hyperpage}{20}
\indexentry{IntoLoop@\texttt {IntoLoop}|hyperpage}{20}
\indexentry{IntoGroup@\texttt {IntoGroup}|hyperpage}{20}
\indexentry{DirectProduct@\texttt {DirectProduct}|hyperpage}{21}
\indexentry{opposite quasigroup|hyperpage}{21}
\indexentry{quasigroup!opposite|hyperpage}{21}
\indexentry{Opposite@\texttt {Opposite}|hyperpage}{21}
\indexentry{OppositeQuasigroup@\texttt {OppositeQuasigroup}|hyperpage}{21}
\indexentry{OppositeLoop@\texttt {OppositeLoop}|hyperpage}{21}
\indexentry{Elements@\texttt {Elements}|hyperpage}{22}
\indexentry{CayleyTable@\texttt {CayleyTable}|hyperpage}{22}
\indexentry{One@\texttt {One}|hyperpage}{22}
\indexentry{Size@\texttt {Size}|hyperpage}{22}
\indexentry{Exponent@\texttt {Exponent}|hyperpage}{23}
\indexentry{loop!power associative|hyperpage}{23}
\indexentry{power associative loop|hyperpage}{23}
\indexentry{exponent|hyperpage}{23}
\indexentry{LeftDivision@\texttt {LeftDivision}|hyperpage}{23}
\indexentry{RightDivision@\texttt {RightDivision}|hyperpage}{23}
\indexentry{LeftDivision@\texttt {LeftDivision}|hyperpage}{23}
\indexentry{LeftDivision@\texttt {LeftDivision}|hyperpage}{23}
\indexentry{RightDivision@\texttt {RightDivision}|hyperpage}{23}
\indexentry{RightDivision@\texttt {RightDivision}|hyperpage}{23}
\indexentry{LeftDivisionCayleyTable@\texttt {LeftDivisionCayleyTable}|hyperpage}{23}
\indexentry{RightDivisionCayleyTable@\texttt {RightDivisionCayleyTable}|hyperpage}{23}
\indexentry{inverse!left|hyperpage}{24}
\indexentry{inverse!right|hyperpage}{24}
\indexentry{inverse|hyperpage}{24}
\indexentry{LeftInverse@\texttt {LeftInverse}|hyperpage}{24}
\indexentry{RightInverse@\texttt {RightInverse}|hyperpage}{24}
\indexentry{Inverse@\texttt {Inverse}|hyperpage}{24}
\indexentry{Associator@\texttt {Associator}|hyperpage}{24}
\indexentry{Commutator@\texttt {Commutator}|hyperpage}{24}
\indexentry{GeneratorsOfQuasigroup@\texttt {GeneratorsOfQuasigroup}|hyperpage}{24}
\indexentry{GeneratorsOfLoop@\texttt {GeneratorsOfLoop}|hyperpage}{24}
\indexentry{GeneratorsSmallest@\texttt {GeneratorsSmallest}|hyperpage}{25}
\indexentry{SmallGeneratingSet@\texttt {SmallGeneratingSet}|hyperpage}{25}
\indexentry{Parent@\texttt {Parent}|hyperpage}{26}
\indexentry{Position@\texttt {Position}|hyperpage}{26}
\indexentry{PosInParent@\texttt {PosInParent}|hyperpage}{27}
\indexentry{Subquasigroup@\texttt {Subquasigroup}|hyperpage}{27}
\indexentry{Subloop@\texttt {Subloop}|hyperpage}{27}
\indexentry{IsSubquasigroup@\texttt {IsSubquasigroup}|hyperpage}{27}
\indexentry{IsSubloop@\texttt {IsSubloop}|hyperpage}{27}
\indexentry{AllSubquasigroups@\texttt {AllSubquasigroups}|hyperpage}{27}
\indexentry{AllSubloops@\texttt {AllSubloops}|hyperpage}{28}
\indexentry{RightCosets@\texttt {RightCosets}|hyperpage}{28}
\indexentry{coset|hyperpage}{28}
\indexentry{RightTransversal@\texttt {RightTransversal}|hyperpage}{28}
\indexentry{transversal|hyperpage}{28}
\indexentry{LeftTranslation@\texttt {LeftTranslation}|hyperpage}{28}
\indexentry{RightTranslation@\texttt {RightTranslation}|hyperpage}{28}
\indexentry{LeftSection@\texttt {LeftSection}|hyperpage}{28}
\indexentry{RightSection@\texttt {RightSection}|hyperpage}{28}
\indexentry{LeftMultiplicationGroup@\texttt {LeftMultiplicationGroup}|hyperpage}{29}
\indexentry{RightMultiplicationGroup@\texttt {RightMultiplicationGroup}|hyperpage}{29}
\indexentry{MultiplicationGroup@\texttt {MultiplicationGroup}|hyperpage}{29}
\indexentry{RelativeLeftMultiplicationGroup@\texttt {RelativeLeftMultiplicationGroup}|hyperpage}{29}
\indexentry{RelativeRightMultiplicationGroup@\texttt {RelativeRightMultiplicationGroup}|hyperpage}{29}
\indexentry{RelativeMultiplicationGroup@\texttt {RelativeMultiplicationGroup}|hyperpage}{29}
\indexentry{multiplication group!relative left|hyperpage}{29}
\indexentry{multiplication group!relative right |hyperpage}{29}
\indexentry{multiplication group!relative|hyperpage}{29}
\indexentry{inner mapping!left|hyperpage}{29}
\indexentry{inner mapping!right|hyperpage}{29}
\indexentry{conjugation|hyperpage}{30}
\indexentry{inner mapping!middle|hyperpage}{30}
\indexentry{inner mapping group!middle|hyperpage}{30}
\indexentry{LeftInnerMapping@\texttt {LeftInnerMapping}|hyperpage}{30}
\indexentry{RightInnerMapping@\texttt {RightInnerMapping}|hyperpage}{30}
\indexentry{MiddleInnerMapping@\texttt {MiddleInnerMapping}|hyperpage}{30}
\indexentry{LeftInnerMappingGroup@\texttt {LeftInnerMappingGroup}|hyperpage}{30}
\indexentry{RightInnerMappingGroup@\texttt {RightInnerMappingGroup}|hyperpage}{30}
\indexentry{MiddleInnerMappingGroup@\texttt {MiddleInnerMappingGroup}|hyperpage}{30}
\indexentry{InnerMappingGroup@\texttt {InnerMappingGroup}|hyperpage}{30}
\indexentry{LeftNucleus@\texttt {LeftNucleus}|hyperpage}{30}
\indexentry{MiddleNucleus@\texttt {MiddleNucleus}|hyperpage}{30}
\indexentry{RightNucleus@\texttt {RightNucleus}|hyperpage}{30}
\indexentry{Nuc@\texttt {Nuc}|hyperpage}{31}
\indexentry{NucleusOfQuasigroup@\texttt {NucleusOfQuasigroup}|hyperpage}{31}
\indexentry{NucleusOfLoop@\texttt {NucleusOfLoop}|hyperpage}{31}
\indexentry{Commutant@\texttt {Commutant}|hyperpage}{31}
\indexentry{Center@\texttt {Center}|hyperpage}{31}
\indexentry{AssociatorSubloop@\texttt {AssociatorSubloop}|hyperpage}{31}
\indexentry{IsNormal@\texttt {IsNormal}|hyperpage}{31}
\indexentry{subloop!normal|hyperpage}{31}
\indexentry{normal subloop|hyperpage}{31}
\indexentry{NormalClosure@\texttt {NormalClosure}|hyperpage}{31}
\indexentry{normal closure|hyperpage}{31}
\indexentry{IsSimple@\texttt {IsSimple}|hyperpage}{32}
\indexentry{simple loop|hyperpage}{32}
\indexentry{loop!simple|hyperpage}{32}
\indexentry{FactorLoop@\texttt {FactorLoop}|hyperpage}{32}
\indexentry{NaturalHomomorphismByNormalSubloop@\texttt {NaturalHomomorphismByNormalSubloop}|hyperpage}{32}
\indexentry{IsNilpotent@\texttt {IsNilpotent}|hyperpage}{32}
\indexentry{NilpotencyClassOfLoop@\texttt {NilpotencyClassOfLoop}|hyperpage}{32}
\indexentry{IsStronglyNilpotent@\texttt {IsStronglyNilpotent}|hyperpage}{32}
\indexentry{strongly nilpotent loop|hyperpage}{32}
\indexentry{nilpotent loop!strongly|hyperpage}{32}
\indexentry{loop!strongly nilpotent|hyperpage}{32}
\indexentry{UpperCentralSeries@\texttt {UpperCentralSeries}|hyperpage}{33}
\indexentry{LowerCentralSeries@\texttt {LowerCentralSeries}|hyperpage}{33}
\indexentry{central series!lower|hyperpage}{33}
\indexentry{IsSolvable@\texttt {IsSolvable}|hyperpage}{33}
\indexentry{DerivedSubloop@\texttt {DerivedSubloop}|hyperpage}{33}
\indexentry{DerivedLength@\texttt {DerivedLength}|hyperpage}{33}
\indexentry{FrattiniSubloop@\texttt {FrattiniSubloop}|hyperpage}{33}
\indexentry{Frattini subloop|hyperpage}{33}
\indexentry{FrattinifactorSize@\texttt {FrattinifactorSize}|hyperpage}{33}
\indexentry{IsomorphismQuasigroups@\texttt {IsomorphismQuasigroups}|hyperpage}{33}
\indexentry{IsomorphismLoops@\texttt {IsomorphismLoops}|hyperpage}{34}
\indexentry{QuasigroupsUpToIsomorphism@\texttt {QuasigroupsUpToIsomorphism}|hyperpage}{34}
\indexentry{LoopsUpToIsomorphism@\texttt {LoopsUpToIsomorphism}|hyperpage}{34}
\indexentry{AutomorphismGroup@\texttt {AutomorphismGroup}|hyperpage}{34}
\indexentry{IsomorphicCopyByPerm@\texttt {IsomorphicCopyByPerm}|hyperpage}{34}
\indexentry{IsomorphicCopyByNormalSubloop@\texttt {IsomorphicCopyByNormalSubloop}|hyperpage}{34}
\indexentry{Discriminator@\texttt {Discriminator}|hyperpage}{35}
\indexentry{AreEqualDiscriminators@\texttt {AreEqualDiscriminators}|hyperpage}{35}
\indexentry{IsotopismLoops@\texttt {IsotopismLoops}|hyperpage}{35}
\indexentry{LoopsUpToIsotopism@\texttt {LoopsUpToIsotopism}|hyperpage}{35}
\indexentry{IsAssociative@\texttt {IsAssociative}|hyperpage}{36}
\indexentry{IsCommutative@\texttt {IsCommutative}|hyperpage}{36}
\indexentry{IsPowerAssociative@\texttt {IsPowerAssociative}|hyperpage}{36}
\indexentry{quasigroup!power associative|hyperpage}{36}
\indexentry{power associative quasigroup|hyperpage}{36}
\indexentry{IsDiassociative@\texttt {IsDiassociative}|hyperpage}{36}
\indexentry{quasigroup!diassociative|hyperpage}{37}
\indexentry{diassociative quasigroup|hyperpage}{37}
\indexentry{inverse!left|hyperpage}{37}
\indexentry{inverse!right|hyperpage}{37}
\indexentry{HasLeftInverseProperty@\texttt {HasLeftInverseProperty}|hyperpage}{37}
\indexentry{HasRightInverseProperty@\texttt {HasRightInverseProperty}|hyperpage}{37}
\indexentry{HasInverseProperty@\texttt {HasInverseProperty}|hyperpage}{37}
\indexentry{inverse property!left|hyperpage}{37}
\indexentry{inverse property!right|hyperpage}{37}
\indexentry{inverse property|hyperpage}{37}
\indexentry{HasTwosidedInverses@\texttt {HasTwosidedInverses}|hyperpage}{37}
\indexentry{inverse!two-sided|hyperpage}{37}
\indexentry{HasWeakInverseProperty@\texttt {HasWeakInverseProperty}|hyperpage}{37}
\indexentry{inverse property!weak|hyperpage}{37}
\indexentry{HasAutomorphicInverseProperty@\texttt {HasAutomorphicInverseProperty}|hyperpage}{37}
\indexentry{automorphic inverse property|hyperpage}{37}
\indexentry{inverse property!automorphic|hyperpage}{37}
\indexentry{HasAntiautomorphicInverseProperty@\texttt {HasAntiautomorphicInverseProperty}|hyperpage}{37}
\indexentry{antiautomorphic inverse property|hyperpage}{37}
\indexentry{inverse property!antiautomorphic|hyperpage}{37}
\indexentry{IsSemisymmetric@\texttt {IsSemisymmetric}|hyperpage}{38}
\indexentry{semisymmetric quasigroup|hyperpage}{38}
\indexentry{quasigroup!semisymmetric|hyperpage}{38}
\indexentry{IsTotallySymmetric@\texttt {IsTotallySymmetric}|hyperpage}{38}
\indexentry{totally symmetric quasigroup|hyperpage}{38}
\indexentry{quasigroup!totally symmetric|hyperpage}{38}
\indexentry{IsIdempotent@\texttt {IsIdempotent}|hyperpage}{38}
\indexentry{idempotent quasigroup|hyperpage}{38}
\indexentry{quasigroup!idempotent|hyperpage}{38}
\indexentry{IsSteinerQuasigroup@\texttt {IsSteinerQuasigroup}|hyperpage}{38}
\indexentry{Steiner quasigroup|hyperpage}{38}
\indexentry{quasigroup!Steiner|hyperpage}{38}
\indexentry{unipotent quasigroup|hyperpage}{38}
\indexentry{quasigroup!unipotent|hyperpage}{38}
\indexentry{IsUnipotent@\texttt {IsUnipotent}|hyperpage}{38}
\indexentry{IsLeftDistributive@\texttt {IsLeftDistributive}|hyperpage}{38}
\indexentry{IsRightDistributive@\texttt {IsRightDistributive}|hyperpage}{38}
\indexentry{IsDistributive@\texttt {IsDistributive}|hyperpage}{38}
\indexentry{quasigroup!left distributive|hyperpage}{38}
\indexentry{distributive quasigroup!left|hyperpage}{38}
\indexentry{quasigroup!right distributive|hyperpage}{38}
\indexentry{distributive quasigroup!right|hyperpage}{38}
\indexentry{quasigroup!distributive|hyperpage}{38}
\indexentry{distributive quasigroup|hyperpage}{38}
\indexentry{IsEntropic@\texttt {IsEntropic}|hyperpage}{39}
\indexentry{IsMedial@\texttt {IsMedial}|hyperpage}{39}
\indexentry{entropic quasigroup|hyperpage}{39}
\indexentry{quasigroup!entropic|hyperpage}{39}
\indexentry{medial quasigroup|hyperpage}{39}
\indexentry{quasigroup!medial|hyperpage}{39}
\indexentry{loop!of Bol-Moufang type|hyperpage}{39}
\indexentry{identity!of Bol-Moufang type|hyperpage}{39}
\indexentry{alternative loop!left|hyperpage}{39}
\indexentry{loop!left alternative|hyperpage}{39}
\indexentry{alternative loop!right|hyperpage}{39}
\indexentry{loop!right alternative|hyperpage}{39}
\indexentry{nuclear square loop!left|hyperpage}{39}
\indexentry{loop!left nuclear square|hyperpage}{39}
\indexentry{nuclear square loop!middle|hyperpage}{39}
\indexentry{loop!middle nuclear square|hyperpage}{39}
\indexentry{nuclear square loop!right|hyperpage}{39}
\indexentry{loop!right nuclear square|hyperpage}{39}
\indexentry{flexible loop|hyperpage}{39}
\indexentry{loop!flexible|hyperpage}{39}
\indexentry{Bol loop!left|hyperpage}{39}
\indexentry{loop!left Bol|hyperpage}{39}
\indexentry{Bol loop!right|hyperpage}{39}
\indexentry{loop!right Bol|hyperpage}{39}
\indexentry{LC loop|hyperpage}{39}
\indexentry{loop!LC|hyperpage}{39}
\indexentry{RC loop|hyperpage}{39}
\indexentry{loop!RC|hyperpage}{39}
\indexentry{Moufang loop|hyperpage}{39}
\indexentry{loop!Moufang|hyperpage}{39}
\indexentry{C loop|hyperpage}{39}
\indexentry{loop!C|hyperpage}{39}
\indexentry{extra loop|hyperpage}{39}
\indexentry{loop!extra|hyperpage}{39}
\indexentry{alternative loop|hyperpage}{39}
\indexentry{loop!alternative|hyperpage}{39}
\indexentry{nuclear square loop|hyperpage}{39}
\indexentry{loop!nuclear square|hyperpage}{39}
\indexentry{IsExtraLoop@\texttt {IsExtraLoop}|hyperpage}{40}
\indexentry{IsMoufangLoop@\texttt {IsMoufangLoop}|hyperpage}{40}
\indexentry{IsCLoop@\texttt {IsCLoop}|hyperpage}{40}
\indexentry{IsLeftBolLoop@\texttt {IsLeftBolLoop}|hyperpage}{40}
\indexentry{IsRightBolLoop@\texttt {IsRightBolLoop}|hyperpage}{40}
\indexentry{IsLCLoop@\texttt {IsLCLoop}|hyperpage}{40}
\indexentry{IsRCLoop@\texttt {IsRCLoop}|hyperpage}{40}
\indexentry{IsLeftNuclearSquareLoop@\texttt {IsLeftNuclearSquareLoop}|hyperpage}{40}
\indexentry{IsMiddleNuclearSquareLoop@\texttt {IsMiddleNuclearSquareLoop}|hyperpage}{40}
\indexentry{IsRightNuclearSquareLoop@\texttt {IsRightNuclearSquareLoop}|hyperpage}{40}
\indexentry{IsNuclearSquareLoop@\texttt {IsNuclearSquareLoop}|hyperpage}{41}
\indexentry{IsFlexible@\texttt {IsFlexible}|hyperpage}{41}
\indexentry{IsLeftAlternative@\texttt {IsLeftAlternative}|hyperpage}{41}
\indexentry{IsRightAlternative@\texttt {IsRightAlternative}|hyperpage}{41}
\indexentry{IsAlternative@\texttt {IsAlternative}|hyperpage}{41}
\indexentry{power alternative loop!left|hyperpage}{42}
\indexentry{loop!left power alternative|hyperpage}{42}
\indexentry{power alternative loop!right|hyperpage}{42}
\indexentry{loop!right power alternative|hyperpage}{42}
\indexentry{power alternative loop|hyperpage}{42}
\indexentry{loop!power alternative|hyperpage}{42}
\indexentry{IsLeftPowerAlternative@\texttt {IsLeftPowerAlternative}|hyperpage}{42}
\indexentry{IsRightPowerAlternative@\texttt {IsRightPowerAlternative}|hyperpage}{42}
\indexentry{IsPowerAlternative@\texttt {IsPowerAlternative}|hyperpage}{42}
\indexentry{conjugacy closed loop!left|hyperpage}{42}
\indexentry{loop!left conjugacy closed|hyperpage}{42}
\indexentry{conjugacy closed loop!right|hyperpage}{42}
\indexentry{loop!right conjugacy closed|hyperpage}{42}
\indexentry{conjugacy closed loop|hyperpage}{42}
\indexentry{loop!conjugacy closed|hyperpage}{42}
\indexentry{IsLCCLoop@\texttt {IsLCCLoop}|hyperpage}{42}
\indexentry{IsLeftConjugacyClosedLoop@\texttt {IsLeftConjugacyClosedLoop}|hyperpage}{42}
\indexentry{IsRCCLoop@\texttt {IsRCCLoop}|hyperpage}{42}
\indexentry{IsRightConjugacyClosedLoop@\texttt {IsRightConjugacyClosedLoop}|hyperpage}{42}
\indexentry{IsCCLoop@\texttt {IsCCLoop}|hyperpage}{42}
\indexentry{IsConjugacyClosedLoop@\texttt {IsConjugacyClosedLoop}|hyperpage}{42}
\indexentry{IsOsbornLoop@\texttt {IsOsbornLoop}|hyperpage}{42}
\indexentry{Osborn loop|hyperpage}{43}
\indexentry{loop!Osborn|hyperpage}{43}
\indexentry{automorphic loop!left|hyperpage}{43}
\indexentry{loop!left automorphic|hyperpage}{43}
\indexentry{automorphic loop!middle|hyperpage}{43}
\indexentry{loop!middle automorphic|hyperpage}{43}
\indexentry{automorphic loop!right|hyperpage}{43}
\indexentry{loop!right automorphic|hyperpage}{43}
\indexentry{automorphic loop|hyperpage}{43}
\indexentry{loop!automorphic|hyperpage}{43}
\indexentry{IsLeftAutomorphicLoop@\texttt {IsLeftAutomorphicLoop}|hyperpage}{43}
\indexentry{IsLeftALoop@\texttt {IsLeftALoop}|hyperpage}{43}
\indexentry{IsMiddleAutomorphicLoop@\texttt {IsMiddleAutomorphicLoop}|hyperpage}{43}
\indexentry{IsMiddleALoop@\texttt {IsMiddleALoop}|hyperpage}{43}
\indexentry{IsRightAutomorphicLoop@\texttt {IsRightAutomorphicLoop}|hyperpage}{44}
\indexentry{IsRightALoop@\texttt {IsRightALoop}|hyperpage}{44}
\indexentry{IsAutomorphicLoop@\texttt {IsAutomorphicLoop}|hyperpage}{44}
\indexentry{IsALoop@\texttt {IsALoop}|hyperpage}{44}
\indexentry{IsCodeLoop@\texttt {IsCodeLoop}|hyperpage}{44}
\indexentry{code loop|hyperpage}{44}
\indexentry{loop!code|hyperpage}{44}
\indexentry{IsSteinerLoop@\texttt {IsSteinerLoop}|hyperpage}{44}
\indexentry{Steiner loop|hyperpage}{44}
\indexentry{loop!Steiner|hyperpage}{44}
\indexentry{IsLeftBruckLoop@\texttt {IsLeftBruckLoop}|hyperpage}{44}
\indexentry{IsLeftKLoop@\texttt {IsLeftKLoop}|hyperpage}{44}
\indexentry{Bruck loop!left|hyperpage}{44}
\indexentry{loop!left Bruck|hyperpage}{44}
\indexentry{K loop!left|hyperpage}{44}
\indexentry{loop!left K|hyperpage}{44}
\indexentry{IsRightBruckLoop@\texttt {IsRightBruckLoop}|hyperpage}{44}
\indexentry{IsRightKLoop@\texttt {IsRightKLoop}|hyperpage}{44}
\indexentry{Bruck loop!right|hyperpage}{44}
\indexentry{loop!right Bruck|hyperpage}{44}
\indexentry{K loop!right|hyperpage}{44}
\indexentry{loop!right K|hyperpage}{44}
\indexentry{AssociatedLeftBruckLoop@\texttt {AssociatedLeftBruckLoop}|hyperpage}{45}
\indexentry{AssociatedRightBruckLoop@\texttt {AssociatedRightBruckLoop}|hyperpage}{45}
\indexentry{loop!left Bol|hyperpage}{45}
\indexentry{Bol loop!left|hyperpage}{45}
\indexentry{Bruck loop!associated left|hyperpage}{45}
\indexentry{loop!associated left Bruck|hyperpage}{45}
\indexentry{IsExactGroupFactorization@\texttt {IsExactGroupFactorization}|hyperpage}{45}
\indexentry{exact group factorization|hyperpage}{45}
\indexentry{RightBolLoopByExactGroupFactorization@\texttt {Right}\discretionary {-}{}{}\texttt {Bol}\discretionary {-}{}{}\texttt {Loop}\discretionary {-}{}{}\texttt {By}\discretionary {-}{}{}\texttt {Exact}\discretionary {-}{}{}\texttt {Group}\discretionary {-}{}{}\texttt {Factorization}|hyperpage}{45}
\indexentry{modification!Moufang|hyperpage}{46}
\indexentry{LoopByCyclicModification@\texttt {LoopByCyclicModification}|hyperpage}{46}
\indexentry{modification!cyclic|hyperpage}{46}
\indexentry{LoopByDihedralModification@\texttt {LoopByDihedralModification}|hyperpage}{46}
\indexentry{modification!dihedral|hyperpage}{46}
\indexentry{LoopMG2@\texttt {LoopMG2}|hyperpage}{46}
\indexentry{Chein loop|hyperpage}{46}
\indexentry{loop!Chein|hyperpage}{46}
\indexentry{group with triality|hyperpage}{46}
\indexentry{TrialityPermGroup@\texttt {TrialityPermGroup}|hyperpage}{47}
\indexentry{TrialityPcGroup@\texttt {TrialityPcGroup}|hyperpage}{47}
\indexentry{AllLoopTablesInGroup@\texttt {AllLoopTablesInGroup}|hyperpage}{47}
\indexentry{AllProperLoopTablesInGroup@\texttt {AllProperLoopTablesInGroup}|hyperpage}{47}
\indexentry{OneLoopTableInGroup@\texttt {OneLoopTableInGroup}|hyperpage}{47}
\indexentry{OneProperLoopTableInGroup@\texttt {OneProperLoopTableInGroup}|hyperpage}{48}
\indexentry{AllLoopsWithMltGroup@\texttt {AllLoopsWithMltGroup}|hyperpage}{48}
\indexentry{OneLoopWithMltGroup@\texttt {OneLoopWithMltGroup}|hyperpage}{48}
\indexentry{LibraryLoop@\texttt {LibraryLoop}|hyperpage}{49}
\indexentry{MyLibraryLoop@\texttt {MyLibraryLoop}|hyperpage}{49}
\indexentry{DisplayLibraryInfo@\texttt {DisplayLibraryInfo}|hyperpage}{50}
\indexentry{LeftBolLoop@\texttt {LeftBolLoop}|hyperpage}{50}
\indexentry{RightBolLoop@\texttt {RightBolLoop}|hyperpage}{50}
\indexentry{MoufangLoop@\texttt {MoufangLoop}|hyperpage}{50}
\indexentry{octonion loop|hyperpage}{50}
\indexentry{loop!octonion|hyperpage}{50}
\indexentry{CodeLoop@\texttt {CodeLoop}|hyperpage}{51}
\indexentry{SteinerLoop@\texttt {SteinerLoop}|hyperpage}{51}
\indexentry{RCCLoop@\texttt {RCCLoop}|hyperpage}{52}
\indexentry{RightConjugacyClosedLoop@\texttt {RightConjugacyClosedLoop}|hyperpage}{52}
\indexentry{LCCLoop@\texttt {LCCLoop}|hyperpage}{52}
\indexentry{LeftConjugacyClosedLoop@\texttt {LeftConjugacyClosedLoop}|hyperpage}{52}
\indexentry{CCLoop@\texttt {CCLoop}|hyperpage}{52}
\indexentry{ConjugacyClosedLoop@\texttt {ConjugacyClosedLoop}|hyperpage}{52}
\indexentry{SmallLoop@\texttt {SmallLoop}|hyperpage}{53}
\indexentry{Paige loop|hyperpage}{53}
\indexentry{loop!Paige|hyperpage}{53}
\indexentry{PaigeLoop@\texttt {PaigeLoop}|hyperpage}{53}
\indexentry{NilpotentLoop@\texttt {NilpotentLoop}|hyperpage}{53}
\indexentry{AutomorphicLoop@\texttt {AutomorphicLoop}|hyperpage}{53}
\indexentry{sedenion loop|hyperpage}{54}
\indexentry{loop!sedenion|hyperpage}{54}
\indexentry{InterestingLoop@\texttt {InterestingLoop}|hyperpage}{54}
\indexentry{ItpSmallLoop@\texttt {ItpSmallLoop}|hyperpage}{54}

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This is makeindex, version 2.15 [MiKTeX 2.9 64-bit] (kpathsea + Thai support).
Scanning input file loops.idx....done (427 entries accepted, 0 rejected).
Sorting entries......done (4056 comparisons).
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\begin{theindex}
\item \texttt {AllLoopsWithMltGroup}, \hyperpage{48}
\item \texttt {AllLoopTablesInGroup}, \hyperpage{47}
\item \texttt {AllProperLoopTablesInGroup}, \hyperpage{47}
\item \texttt {AllSubloops}, \hyperpage{28}
\item \texttt {AllSubquasigroups}, \hyperpage{27}
\item alternative loop, \hyperpage{39}
\subitem left, \hyperpage{39}
\subitem right, \hyperpage{39}
\item antiautomorphic inverse property, \hyperpage{37}
\item \texttt {AreEqualDiscriminators}, \hyperpage{35}
\item \texttt {AssociatedLeftBruckLoop}, \hyperpage{45}
\item \texttt {AssociatedRightBruckLoop}, \hyperpage{45}
\item \texttt {Associator}, \hyperpage{24}
\item associator, \hyperpage{9}
\item associator subloop, \hyperpage{9}
\item \texttt {AssociatorSubloop}, \hyperpage{31}
\item automorphic inverse property, \hyperpage{37}
\item automorphic loop, \hyperpage{43}
\subitem left, \hyperpage{43}
\subitem middle, \hyperpage{43}
\subitem right, \hyperpage{43}
\item \texttt {AutomorphicLoop}, \hyperpage{53}
\item \texttt {AutomorphismGroup}, \hyperpage{34}
\indexspace
\item Bol loop
\subitem left, \hyperpage{12}, \hyperpage{39}, \hyperpage{45}
\subitem right, \hyperpage{39}
\item Bruck loop
\subitem associated left, \hyperpage{45}
\subitem left, \hyperpage{44}
\subitem right, \hyperpage{44}
\indexspace
\item C loop, \hyperpage{39}
\item \texttt {CanonicalCayleyTable}, \hyperpage{15}
\item \texttt {CanonicalCopy}, \hyperpage{15}
\item Cayley table, \hyperpage{14}
\subitem canonical, \hyperpage{15}
\item \texttt {CayleyTable}, \hyperpage{22}
\item \texttt {CayleyTableByPerms}, \hyperpage{17}
\item \texttt {CCLoop}, \hyperpage{52}
\item \texttt {Center}, \hyperpage{31}
\item center, \hyperpage{9}
\item central series
\subitem lower, \hyperpage{33}
\subitem upper, \hyperpage{9}
\item Chein loop, \hyperpage{46}
\item cocycle, \hyperpage{18}
\item code loop, \hyperpage{44}
\item \texttt {CodeLoop}, \hyperpage{51}
\item \texttt {Commutant}, \hyperpage{31}
\item commutant, \hyperpage{9}
\item \texttt {Commutator}, \hyperpage{24}
\item commutator, \hyperpage{9}
\item conjugacy closed loop, \hyperpage{42}
\subitem left, \hyperpage{42}
\subitem right, \hyperpage{42}
\item \texttt {ConjugacyClosedLoop}, \hyperpage{52}
\item conjugation, \hyperpage{30}
\item coset, \hyperpage{28}
\indexspace
\item derived series, \hyperpage{9}
\item derived subloop, \hyperpage{9}
\item \texttt {DerivedLength}, \hyperpage{33}
\item \texttt {DerivedSubloop}, \hyperpage{33}
\item diassociative quasigroup, \hyperpage{37}
\item \texttt {DirectProduct}, \hyperpage{21}
\item \texttt {Discriminator}, \hyperpage{35}
\item \texttt {DisplayLibraryInfo}, \hyperpage{50}
\item distributive quasigroup, \hyperpage{38}
\subitem left, \hyperpage{38}
\subitem right, \hyperpage{38}
\item division
\subitem left, \hyperpage{8}
\subitem right, \hyperpage{8}
\indexspace
\item \texttt {Elements}, \hyperpage{22}
\item entropic quasigroup, \hyperpage{39}
\item exact group factorization, \hyperpage{45}
\item \texttt {Exponent}, \hyperpage{23}
\item exponent, \hyperpage{23}
\item extension, \hyperpage{18}
\subitem nuclear, \hyperpage{18}
\item extra loop, \hyperpage{39}
\indexspace
\item \texttt {FactorLoop}, \hyperpage{32}
\item flexible loop, \hyperpage{39}
\item folder
\subitem quasigroup, \hyperpage{18}
\item Frattini subloop, \hyperpage{33}
\item \texttt {FrattinifactorSize}, \hyperpage{33}
\item \texttt {FrattiniSubloop}, \hyperpage{33}
\indexspace
\item \texttt {GeneratorsOfLoop}, \hyperpage{24}
\item \texttt {GeneratorsOfQuasigroup}, \hyperpage{24}
\item \texttt {GeneratorsSmallest}, \hyperpage{25}
\item group, \hyperpage{8}
\item group with triality, \hyperpage{46}
\item groupoid, \hyperpage{8}
\indexspace
\item \texttt {HasAntiautomorphicInverseProperty}, \hyperpage{37}
\item \texttt {HasAutomorphicInverseProperty}, \hyperpage{37}
\item \texttt {HasInverseProperty}, \hyperpage{37}
\item \texttt {HasLeftInverseProperty}, \hyperpage{37}
\item \texttt {HasRightInverseProperty}, \hyperpage{37}
\item \texttt {HasTwosidedInverses}, \hyperpage{37}
\item \texttt {HasWeakInverseProperty}, \hyperpage{37}
\item homomorphism, \hyperpage{9}
\item homotopism, \hyperpage{10}
\indexspace
\item idempotent quasigroup, \hyperpage{38}
\item identity
\subitem element, \hyperpage{8}
\subitem of Bol-Moufang type, \hyperpage{39}
\item inner mapping
\subitem left, \hyperpage{29}
\subitem middle, \hyperpage{30}
\subitem right, \hyperpage{29}
\item inner mapping group, \hyperpage{9}
\subitem left, \hyperpage{9}
\subitem middle, \hyperpage{30}
\subitem right, \hyperpage{9}
\item \texttt {InnerMappingGroup}, \hyperpage{30}
\item \texttt {InterestingLoop}, \hyperpage{54}
\item \texttt {IntoGroup}, \hyperpage{20}
\item \texttt {IntoLoop}, \hyperpage{20}
\item \texttt {IntoQuasigroup}, \hyperpage{20}
\item \texttt {Inverse}, \hyperpage{24}
\item inverse, \hyperpage{24}
\subitem left, \hyperpage{24}, \hyperpage{37}
\subitem right, \hyperpage{24}, \hyperpage{37}
\subitem two-sided, \hyperpage{8}, \hyperpage{37}
\item inverse property, \hyperpage{37}
\subitem antiautomorphic, \hyperpage{37}
\subitem automorphic, \hyperpage{37}
\subitem left, \hyperpage{37}
\subitem right, \hyperpage{37}
\subitem weak, \hyperpage{37}
\item \texttt {IsALoop}, \hyperpage{44}
\item \texttt {IsAlternative}, \hyperpage{41}
\item \texttt {IsAssociative}, \hyperpage{36}
\item \texttt {IsAutomorphicLoop}, \hyperpage{44}
\item \texttt {IsCCLoop}, \hyperpage{42}
\item \texttt {IsCLoop}, \hyperpage{40}
\item \texttt {IsCodeLoop}, \hyperpage{44}
\item \texttt {IsCommutative}, \hyperpage{36}
\item \texttt {IsConjugacyClosedLoop}, \hyperpage{42}
\item \texttt {IsDiassociative}, \hyperpage{36}
\item \texttt {IsDistributive}, \hyperpage{38}
\item \texttt {IsEntropic}, \hyperpage{39}
\item \texttt {IsExactGroupFactorization}, \hyperpage{45}
\item \texttt {IsExtraLoop}, \hyperpage{40}
\item \texttt {IsFlexible}, \hyperpage{41}
\item \texttt {IsIdempotent}, \hyperpage{38}
\item \texttt {IsLCCLoop}, \hyperpage{42}
\item \texttt {IsLCLoop}, \hyperpage{40}
\item \texttt {IsLeftALoop}, \hyperpage{43}
\item \texttt {IsLeftAlternative}, \hyperpage{41}
\item \texttt {IsLeftAutomorphicLoop}, \hyperpage{43}
\item \texttt {IsLeftBolLoop}, \hyperpage{40}
\item \texttt {IsLeftBruckLoop}, \hyperpage{44}
\item \texttt {IsLeftConjugacyClosedLoop}, \hyperpage{42}
\item \texttt {IsLeftDistributive}, \hyperpage{38}
\item \texttt {IsLeftKLoop}, \hyperpage{44}
\item \texttt {IsLeftNuclearSquareLoop}, \hyperpage{40}
\item \texttt {IsLeftPowerAlternative}, \hyperpage{42}
\item IsLoop, \hyperpage{11}
\item \texttt {IsLoopCayleyTable}, \hyperpage{14}
\item IsLoopElement, \hyperpage{11}
\item \texttt {IsLoopTable}, \hyperpage{14}
\item \texttt {IsMedial}, \hyperpage{39}
\item \texttt {IsMiddleALoop}, \hyperpage{43}
\item \texttt {IsMiddleAutomorphicLoop}, \hyperpage{43}
\item \texttt {IsMiddleNuclearSquareLoop}, \hyperpage{40}
\item \texttt {IsMoufangLoop}, \hyperpage{40}
\item \texttt {IsNilpotent}, \hyperpage{32}
\item \texttt {IsNormal}, \hyperpage{31}
\item \texttt {IsNuclearSquareLoop}, \hyperpage{41}
\item \texttt {IsomorphicCopyByNormalSubloop}, \hyperpage{34}
\item \texttt {IsomorphicCopyByPerm}, \hyperpage{34}
\item isomorphism, \hyperpage{9}
\item \texttt {IsomorphismLoops}, \hyperpage{34}
\item \texttt {IsomorphismQuasigroups}, \hyperpage{33}
\item \texttt {IsOsbornLoop}, \hyperpage{42}
\item isotopism, \hyperpage{10}
\subitem principal, \hyperpage{10}
\item \texttt {IsotopismLoops}, \hyperpage{35}
\item \texttt {IsPowerAlternative}, \hyperpage{42}
\item \texttt {IsPowerAssociative}, \hyperpage{36}
\item IsQuasigroup, \hyperpage{11}
\item \texttt {IsQuasigroupCayleyTable}, \hyperpage{14}
\item IsQuasigroupElement, \hyperpage{11}
\item \texttt {IsQuasigroupTable}, \hyperpage{14}
\item \texttt {IsRCCLoop}, \hyperpage{42}
\item \texttt {IsRCLoop}, \hyperpage{40}
\item \texttt {IsRightALoop}, \hyperpage{44}
\item \texttt {IsRightAlternative}, \hyperpage{41}
\item \texttt {IsRightAutomorphicLoop}, \hyperpage{44}
\item \texttt {IsRightBolLoop}, \hyperpage{40}
\item \texttt {IsRightBruckLoop}, \hyperpage{44}
\item \texttt {IsRightConjugacyClosedLoop}, \hyperpage{42}
\item \texttt {IsRightDistributive}, \hyperpage{38}
\item \texttt {IsRightKLoop}, \hyperpage{44}
\item \texttt {IsRightNuclearSquareLoop}, \hyperpage{40}
\item \texttt {IsRightPowerAlternative}, \hyperpage{42}
\item \texttt {IsSemisymmetric}, \hyperpage{38}
\item \texttt {IsSimple}, \hyperpage{32}
\item \texttt {IsSolvable}, \hyperpage{33}
\item \texttt {IsSteinerLoop}, \hyperpage{44}
\item \texttt {IsSteinerQuasigroup}, \hyperpage{38}
\item \texttt {IsStronglyNilpotent}, \hyperpage{32}
\item \texttt {IsSubloop}, \hyperpage{27}
\item \texttt {IsSubquasigroup}, \hyperpage{27}
\item \texttt {IsTotallySymmetric}, \hyperpage{38}
\item \texttt {IsUnipotent}, \hyperpage{38}
\item \texttt {ItpSmallLoop}, \hyperpage{54}
\indexspace
\item K loop
\subitem left, \hyperpage{44}
\subitem right, \hyperpage{44}
\indexspace
\item latin square, \hyperpage{8}, \hyperpage{14}
\subitem random, \hyperpage{19}
\item LC loop, \hyperpage{39}
\item \texttt {LCCLoop}, \hyperpage{52}
\item \texttt {LeftBolLoop}, \hyperpage{50}
\item \texttt {LeftConjugacyClosedLoop}, \hyperpage{52}
\item \texttt {LeftDivision}, \hyperpage{23}
\item \texttt {LeftDivisionCayleyTable}, \hyperpage{23}
\item \texttt {LeftInnerMapping}, \hyperpage{30}
\item \texttt {LeftInnerMappingGroup}, \hyperpage{30}
\item \texttt {LeftInverse}, \hyperpage{24}
\item \texttt {LeftMultiplicationGroup}, \hyperpage{29}
\item \texttt {LeftNucleus}, \hyperpage{30}
\item \texttt {LeftSection}, \hyperpage{28}
\item \texttt {LeftTranslation}, \hyperpage{28}
\item \texttt {LibraryLoop}, \hyperpage{49}
\item loop, \hyperpage{8}
\subitem alternative, \hyperpage{39}
\subitem associated left Bruck, \hyperpage{45}
\subitem automorphic, \hyperpage{43}
\subitem C, \hyperpage{39}
\subitem Chein, \hyperpage{46}
\subitem code, \hyperpage{44}
\subitem conjugacy closed, \hyperpage{42}
\subitem extra, \hyperpage{39}
\subitem flexible, \hyperpage{39}
\subitem LC, \hyperpage{39}
\subitem left alternative, \hyperpage{39}
\subitem left automorphic, \hyperpage{43}
\subitem left Bol, \hyperpage{12}, \hyperpage{39}, \hyperpage{45}
\subitem left Bruck, \hyperpage{44}
\subitem left conjugacy closed, \hyperpage{42}
\subitem left K, \hyperpage{44}
\subitem left nuclear square, \hyperpage{39}
\subitem left power alternative, \hyperpage{42}
\subitem middle automorphic, \hyperpage{43}
\subitem middle nuclear square, \hyperpage{39}
\subitem Moufang, \hyperpage{39}
\subitem nilpotent, \hyperpage{9}, \hyperpage{19}
\subitem nuclear square, \hyperpage{39}
\subitem octonion, \hyperpage{50}
\subitem of Bol-Moufang type, \hyperpage{39}
\subitem Osborn, \hyperpage{43}
\subitem Paige, \hyperpage{53}
\subitem power alternative, \hyperpage{42}
\subitem power associative, \hyperpage{23}
\subitem RC, \hyperpage{39}
\subitem right alternative, \hyperpage{39}
\subitem right automorphic, \hyperpage{43}
\subitem right Bol, \hyperpage{39}
\subitem right Bruck, \hyperpage{44}
\subitem right conjugacy closed, \hyperpage{42}
\subitem right K, \hyperpage{44}
\subitem right nuclear square, \hyperpage{39}
\subitem right power alternative, \hyperpage{42}
\subitem sedenion, \hyperpage{54}
\subitem simple, \hyperpage{12}, \hyperpage{32}
\subitem solvable, \hyperpage{9}
\subitem Steiner, \hyperpage{44}
\subitem strongly nilpotent, \hyperpage{32}
\item loop isotope
\subitem principal, \hyperpage{10}
\item loop table, \hyperpage{14}
\item \texttt {LoopByCayleyTable}, \hyperpage{15}
\item \texttt {LoopByCyclicModification}, \hyperpage{46}
\item \texttt {LoopByDihedralModification}, \hyperpage{46}
\item \texttt {LoopByExtension}, \hyperpage{18}
\item \texttt {LoopByLeftSection}, \hyperpage{17}
\item \texttt {LoopByRightFolder}, \hyperpage{18}
\item \texttt {LoopByRightSection}, \hyperpage{17}
\item \texttt {LoopFromFile}, \hyperpage{17}
\item \texttt {LoopMG2}, \hyperpage{46}
\item \texttt {LoopsUpToIsomorphism}, \hyperpage{34}
\item \texttt {LoopsUpToIsotopism}, \hyperpage{35}
\item \texttt {LowerCentralSeries}, \hyperpage{33}
\indexspace
\item magma, \hyperpage{8}
\item medial quasigroup, \hyperpage{39}
\item \texttt {MiddleInnerMapping}, \hyperpage{30}
\item \texttt {MiddleInnerMappingGroup}, \hyperpage{30}
\item \texttt {MiddleNucleus}, \hyperpage{30}
\item modification
\subitem cyclic, \hyperpage{46}
\subitem dihedral, \hyperpage{46}
\subitem Moufang, \hyperpage{46}
\item Moufang loop, \hyperpage{39}
\item \texttt {MoufangLoop}, \hyperpage{50}
\item multiplication group, \hyperpage{9}
\subitem left, \hyperpage{9}
\subitem relative, \hyperpage{29}
\subitem relative left, \hyperpage{29}
\subitem relative right , \hyperpage{29}
\subitem right, \hyperpage{9}
\item multiplication table, \hyperpage{14}
\item \texttt {MultiplicationGroup}, \hyperpage{29}
\item \texttt {MyLibraryLoop}, \hyperpage{49}
\indexspace
\item \texttt {NaturalHomomorphismByNormalSubloop}, \hyperpage{32}
\item neutral element, \hyperpage{8}
\item nilpotence class, \hyperpage{9}
\item \texttt {NilpotencyClassOfLoop}, \hyperpage{32}
\item nilpotent loop, \hyperpage{9}
\subitem strongly, \hyperpage{32}
\item \texttt {NilpotentLoop}, \hyperpage{53}
\item normal closure, \hyperpage{31}
\item normal subloop, \hyperpage{31}
\item \texttt {NormalClosure}, \hyperpage{31}
\item \texttt {NormalizedQuasigroupTable}, \hyperpage{15}
\item \texttt {Nuc}, \hyperpage{31}
\item nuclear square loop, \hyperpage{39}
\subitem left, \hyperpage{39}
\subitem middle, \hyperpage{39}
\subitem right, \hyperpage{39}
\item \texttt {NuclearExtension}, \hyperpage{18}
\item nucleus, \hyperpage{9}
\subitem left, \hyperpage{9}
\subitem middle, \hyperpage{9}
\subitem right, \hyperpage{9}
\item \texttt {NucleusOfLoop}, \hyperpage{31}
\item \texttt {NucleusOfQuasigroup}, \hyperpage{31}
\indexspace
\item octonion loop, \hyperpage{50}
\item \texttt {One}, \hyperpage{22}
\item \texttt {OneLoopTableInGroup}, \hyperpage{47}
\item \texttt {OneLoopWithMltGroup}, \hyperpage{48}
\item \texttt {OneProperLoopTableInGroup}, \hyperpage{48}
\item \texttt {Opposite}, \hyperpage{21}
\item opposite quasigroup, \hyperpage{21}
\item \texttt {OppositeLoop}, \hyperpage{21}
\item \texttt {OppositeQuasigroup}, \hyperpage{21}
\item Osborn loop, \hyperpage{43}
\indexspace
\item Paige loop, \hyperpage{53}
\item \texttt {PaigeLoop}, \hyperpage{53}
\item \texttt {Parent}, \hyperpage{26}
\item \texttt {PosInParent}, \hyperpage{27}
\item \texttt {Position}, \hyperpage{26}
\item power alternative loop, \hyperpage{42}
\subitem left, \hyperpage{42}
\subitem right, \hyperpage{42}
\item power associative loop, \hyperpage{23}
\item power associative quasigroup, \hyperpage{36}
\item \texttt {PrincipalLoopIsotope}, \hyperpage{20}
\indexspace
\item quasigroup, \hyperpage{8}
\subitem diassociative, \hyperpage{37}
\subitem distributive, \hyperpage{38}
\subitem entropic, \hyperpage{39}
\subitem idempotent, \hyperpage{38}
\subitem left distributive, \hyperpage{38}
\subitem medial, \hyperpage{39}
\subitem opposite, \hyperpage{21}
\subitem power associative, \hyperpage{36}
\subitem right distributive, \hyperpage{38}
\subitem semisymmetric, \hyperpage{38}
\subitem Steiner, \hyperpage{38}
\subitem totally symmetric, \hyperpage{38}
\subitem unipotent, \hyperpage{38}
\item quasigroup table, \hyperpage{14}
\item \texttt {QuasigroupByCayleyTable}, \hyperpage{15}
\item \texttt {QuasigroupByLeftSection}, \hyperpage{17}
\item \texttt {QuasigroupByRightFolder}, \hyperpage{18}
\item \texttt {QuasigroupByRightSection}, \hyperpage{17}
\item \texttt {QuasigroupFromFile}, \hyperpage{17}
\item \texttt {QuasigroupsUpToIsomorphism}, \hyperpage{34}
\indexspace
\item \texttt {RandomLoop}, \hyperpage{19}
\item \texttt {RandomNilpotentLoop}, \hyperpage{19}
\item \texttt {RandomQuasigroup}, \hyperpage{19}
\item RC loop, \hyperpage{39}
\item \texttt {RCCLoop}, \hyperpage{52}
\item \texttt {RelativeLeftMultiplicationGroup}, \hyperpage{29}
\item \texttt {RelativeMultiplicationGroup}, \hyperpage{29}
\item \texttt {RelativeRightMultiplicationGroup}, \hyperpage{29}
\item \texttt {RightBolLoop}, \hyperpage{50}
\item \texttt {Right}\discretionary {-}{}{}\texttt {Bol}\discretionary {-}{}{}\texttt {Loop}\discretionary {-}{}{}\texttt {By}\discretionary {-}{}{}\texttt {Exact}\discretionary {-}{}{}\texttt {Group}\discretionary {-}{}{}\texttt {Factorization},
\hyperpage{45}
\item \texttt {RightConjugacyClosedLoop}, \hyperpage{52}
\item \texttt {RightCosets}, \hyperpage{28}
\item \texttt {RightDivision}, \hyperpage{23}
\item \texttt {RightDivisionCayleyTable}, \hyperpage{23}
\item \texttt {RightInnerMapping}, \hyperpage{30}
\item \texttt {RightInnerMappingGroup}, \hyperpage{30}
\item \texttt {RightInverse}, \hyperpage{24}
\item \texttt {RightMultiplicationGroup}, \hyperpage{29}
\item \texttt {RightNucleus}, \hyperpage{30}
\item \texttt {RightSection}, \hyperpage{28}
\item \texttt {RightTranslation}, \hyperpage{28}
\item \texttt {RightTransversal}, \hyperpage{28}
\indexspace
\item section
\subitem left, \hyperpage{8}
\subitem right, \hyperpage{8}
\item sedenion loop, \hyperpage{54}
\item semisymmetric quasigroup, \hyperpage{38}
\item \texttt {SetLoopElmName}, \hyperpage{13}
\item \texttt {SetQuasigroupElmName}, \hyperpage{13}
\item simple loop, \hyperpage{12}, \hyperpage{32}
\item \texttt {Size}, \hyperpage{22}
\item \texttt {SmallGeneratingSet}, \hyperpage{25}
\item \texttt {SmallLoop}, \hyperpage{53}
\item solvability class, \hyperpage{9}
\item solvable loop, \hyperpage{9}
\item Steiner loop, \hyperpage{44}
\item Steiner quasigroup, \hyperpage{38}
\item \texttt {SteinerLoop}, \hyperpage{51}
\item strongly nilpotent loop, \hyperpage{32}
\item \texttt {Subloop}, \hyperpage{27}
\item subloop, \hyperpage{9}
\subitem normal, \hyperpage{9}, \hyperpage{31}
\item \texttt {Subquasigroup}, \hyperpage{27}
\item subquasigroup, \hyperpage{9}
\indexspace
\item totally symmetric quasigroup, \hyperpage{38}
\item translation
\subitem left, \hyperpage{8}
\subitem right, \hyperpage{8}
\item transversal, \hyperpage{28}
\item \texttt {TrialityPcGroup}, \hyperpage{47}
\item \texttt {TrialityPermGroup}, \hyperpage{47}
\indexspace
\item unipotent quasigroup, \hyperpage{38}
\item \texttt {UpperCentralSeries}, \hyperpage{33}
\end{theindex}

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Chapter 1.
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Chapter 2.
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Chapter 3.
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[][]\T1/ptm/m/n/10.95 In some spe-cial sit-u-a-tions, the func-tion \T1/cmtt/m/
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Chapter 4.
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[]\T1/ptm/b/n/10.95 Returns: \T1/ptm/m/n/10.95 If [] is a de-clared magma that
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Chapter 5.
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Chapter 6.
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Chapter 7.
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Chapter 9.
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Appendix B.
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\BOOKMARK [0][-]{chapter.1}{Introduction}{}% 1
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\BOOKMARK [1][-]{section.1.2}{Installation}{chapter.1}% 3
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\BOOKMARK [1][-]{section.1.7}{Acknowledgment}{chapter.1}% 8
\BOOKMARK [0][-]{chapter.2}{Mathematical Background}{}% 9
\BOOKMARK [1][-]{section.2.1}{Quasigroups and Loops}{chapter.2}% 10
\BOOKMARK [1][-]{section.2.2}{Translations}{chapter.2}% 11
\BOOKMARK [1][-]{section.2.3}{Subquasigroups and Subloops}{chapter.2}% 12
\BOOKMARK [1][-]{section.2.4}{Nilpotence and Solvability}{chapter.2}% 13
\BOOKMARK [1][-]{section.2.5}{Associators and Commutators}{chapter.2}% 14
\BOOKMARK [1][-]{section.2.6}{Homomorphism and Homotopisms}{chapter.2}% 15
\BOOKMARK [0][-]{chapter.3}{How the Package Works}{}% 16
\BOOKMARK [1][-]{section.3.1}{Representing Quasigroups}{chapter.3}% 17
\BOOKMARK [1][-]{section.3.2}{Conversions between magmas, quasigroups, loops and groups}{chapter.3}% 18
\BOOKMARK [1][-]{section.3.3}{Calculating with Quasigroups}{chapter.3}% 19
\BOOKMARK [1][-]{section.3.4}{Naming, Viewing and Printing Quasigroups and their Elements}{chapter.3}% 20
\BOOKMARK [0][-]{chapter.4}{Creating Quasigroups and Loops}{}% 21
\BOOKMARK [1][-]{section.4.1}{About Cayley Tables}{chapter.4}% 22
\BOOKMARK [1][-]{section.4.2}{Testing Cayley Tables}{chapter.4}% 23
\BOOKMARK [1][-]{section.4.3}{Canonical and Normalized Cayley Tables}{chapter.4}% 24
\BOOKMARK [1][-]{section.4.4}{Creating Quasigroups and Loops From Cayley Tables}{chapter.4}% 25
\BOOKMARK [1][-]{section.4.5}{Creating Quasigroups and Loops from a File}{chapter.4}% 26
\BOOKMARK [1][-]{section.4.6}{Creating Quasigroups and Loops From Sections}{chapter.4}% 27
\BOOKMARK [1][-]{section.4.7}{Creating Quasigroups and Loops From Folders}{chapter.4}% 28
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\BOOKMARK [1][-]{section.4.10}{Conversions}{chapter.4}% 31
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\BOOKMARK [0][-]{chapter.5}{Basic Methods And Attributes}{}% 34
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\BOOKMARK [1][-]{section.5.2}{Basic Arithmetic Operations}{chapter.5}% 36
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\BOOKMARK [0][-]{chapter.6}{Methods Based on Permutation Groups}{}% 40
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\BOOKMARK [1][-]{section.6.6}{Nuclei, Commutant, Center, and Associator Subloop}{chapter.6}% 46
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\BOOKMARK [1][-]{section.6.8}{Factor Loops}{chapter.6}% 48
\BOOKMARK [1][-]{section.6.9}{Nilpotency and Central Series}{chapter.6}% 49
\BOOKMARK [1][-]{section.6.10}{Solvability, Derived Series and Frattini Subloop}{chapter.6}% 50
\BOOKMARK [1][-]{section.6.11}{Isomorphisms and Automorphisms}{chapter.6}% 51
\BOOKMARK [1][-]{section.6.12}{Isotopisms}{chapter.6}% 52
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\BOOKMARK [1][-]{section.7.3}{Some Properties of Quasigroups}{chapter.7}% 56
\BOOKMARK [1][-]{section.7.4}{Loops of Bol Moufang Type}{chapter.7}% 57
\BOOKMARK [1][-]{section.7.5}{Power Alternative Loops}{chapter.7}% 58
\BOOKMARK [1][-]{section.7.6}{Conjugacy Closed Loops and Related Properties}{chapter.7}% 59
\BOOKMARK [1][-]{section.7.7}{Automorphic Loops}{chapter.7}% 60
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\BOOKMARK [0][-]{chapter.8}{Specific Methods}{}% 62
\BOOKMARK [1][-]{section.8.1}{Core Methods for Bol Loops}{chapter.8}% 63
\BOOKMARK [1][-]{section.8.2}{Moufang Modifications}{chapter.8}% 64
\BOOKMARK [1][-]{section.8.3}{Triality for Moufang Loops}{chapter.8}% 65
\BOOKMARK [1][-]{section.8.4}{Realizing Groups as Multiplication Groups of Loops}{chapter.8}% 66
\BOOKMARK [0][-]{chapter.9}{Libraries of Loops}{}% 67
\BOOKMARK [1][-]{section.9.1}{A Typical Library}{chapter.9}% 68
\BOOKMARK [1][-]{section.9.2}{Left Bol Loops and Right Bol Loops}{chapter.9}% 69
\BOOKMARK [1][-]{section.9.3}{Moufang Loops}{chapter.9}% 70
\BOOKMARK [1][-]{section.9.4}{Code Loops}{chapter.9}% 71
\BOOKMARK [1][-]{section.9.5}{Steiner Loops}{chapter.9}% 72
\BOOKMARK [1][-]{section.9.6}{Conjugacy Closed Loops}{chapter.9}% 73
\BOOKMARK [1][-]{section.9.7}{Small Loops}{chapter.9}% 74
\BOOKMARK [1][-]{section.9.8}{Paige Loops}{chapter.9}% 75
\BOOKMARK [1][-]{section.9.9}{Nilpotent Loops}{chapter.9}% 76
\BOOKMARK [1][-]{section.9.10}{Automorphic Loops}{chapter.9}% 77
\BOOKMARK [1][-]{section.9.11}{Interesting Loops}{chapter.9}% 78
\BOOKMARK [1][-]{section.9.12}{Libraries of Loops Up To Isotopism}{chapter.9}% 79
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\BOOKMARK [0][-]{appendix.B}{Filters}{}% 81
\BOOKMARK [0][-]{appendix*.3}{References}{}% 82
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\contentsline {subsection}{\numberline {4.2.1}\leavevmode {\color {Chapter }IsQuasigroupTable and IsQuasigroupCayleyTable}}{14}{subsection.4.2.1}
\contentsline {subsection}{\numberline {4.2.2}\leavevmode {\color {Chapter }IsLoopTable and IsLoopCayleyTable}}{14}{subsection.4.2.2}
\contentsline {section}{\numberline {4.3}\leavevmode {\color {Chapter }Canonical and Normalized Cayley Tables}}{15}{section.4.3}
\contentsline {subsection}{\numberline {4.3.1}\leavevmode {\color {Chapter }CanonicalCayleyTable}}{15}{subsection.4.3.1}
\contentsline {subsection}{\numberline {4.3.2}\leavevmode {\color {Chapter }CanonicalCopy}}{15}{subsection.4.3.2}
\contentsline {subsection}{\numberline {4.3.3}\leavevmode {\color {Chapter }NormalizedQuasigroupTable}}{15}{subsection.4.3.3}
\contentsline {section}{\numberline {4.4}\leavevmode {\color {Chapter }Creating Quasigroups and Loops From Cayley Tables}}{15}{section.4.4}
\contentsline {subsection}{\numberline {4.4.1}\leavevmode {\color {Chapter }QuasigroupByCayleyTable and LoopByCayleyTable}}{15}{subsection.4.4.1}
\contentsline {section}{\numberline {4.5}\leavevmode {\color {Chapter }Creating Quasigroups and Loops from a File}}{16}{section.4.5}
\contentsline {subsection}{\numberline {4.5.1}\leavevmode {\color {Chapter }QuasigroupFromFile and LoopFromFile}}{17}{subsection.4.5.1}
\contentsline {section}{\numberline {4.6}\leavevmode {\color {Chapter }Creating Quasigroups and Loops From Sections}}{17}{section.4.6}
\contentsline {subsection}{\numberline {4.6.1}\leavevmode {\color {Chapter }CayleyTableByPerms}}{17}{subsection.4.6.1}
\contentsline {subsection}{\numberline {4.6.2}\leavevmode {\color {Chapter }QuasigroupByLeftSection and LoopByLeftSection}}{17}{subsection.4.6.2}
\contentsline {subsection}{\numberline {4.6.3}\leavevmode {\color {Chapter }QuasigroupByRightSection and LoopByRightSection}}{17}{subsection.4.6.3}
\contentsline {section}{\numberline {4.7}\leavevmode {\color {Chapter }Creating Quasigroups and Loops From Folders}}{18}{section.4.7}
\contentsline {subsection}{\numberline {4.7.1}\leavevmode {\color {Chapter }QuasigroupByRightFolder and LoopByRightFolder}}{18}{subsection.4.7.1}
\contentsline {section}{\numberline {4.8}\leavevmode {\color {Chapter }Creating Quasigroups and Loops By Nuclear Extensions}}{18}{section.4.8}
\contentsline {subsection}{\numberline {4.8.1}\leavevmode {\color {Chapter }NuclearExtension}}{18}{subsection.4.8.1}
\contentsline {subsection}{\numberline {4.8.2}\leavevmode {\color {Chapter }LoopByExtension}}{18}{subsection.4.8.2}
\contentsline {section}{\numberline {4.9}\leavevmode {\color {Chapter }Random Quasigroups and Loops}}{19}{section.4.9}
\contentsline {subsection}{\numberline {4.9.1}\leavevmode {\color {Chapter }RandomQuasigroup and RandomLoop}}{19}{subsection.4.9.1}
\contentsline {subsection}{\numberline {4.9.2}\leavevmode {\color {Chapter }RandomNilpotentLoop}}{19}{subsection.4.9.2}
\contentsline {section}{\numberline {4.10}\leavevmode {\color {Chapter }Conversions}}{20}{section.4.10}
\contentsline {subsection}{\numberline {4.10.1}\leavevmode {\color {Chapter }IntoQuasigroup}}{20}{subsection.4.10.1}
\contentsline {subsection}{\numberline {4.10.2}\leavevmode {\color {Chapter }PrincipalLoopIsotope}}{20}{subsection.4.10.2}
\contentsline {subsection}{\numberline {4.10.3}\leavevmode {\color {Chapter }IntoLoop}}{20}{subsection.4.10.3}
\contentsline {subsection}{\numberline {4.10.4}\leavevmode {\color {Chapter }IntoGroup}}{20}{subsection.4.10.4}
\contentsline {section}{\numberline {4.11}\leavevmode {\color {Chapter }Products of Quasigroups and Loops}}{21}{section.4.11}
\contentsline {subsection}{\numberline {4.11.1}\leavevmode {\color {Chapter }DirectProduct}}{21}{subsection.4.11.1}
\contentsline {section}{\numberline {4.12}\leavevmode {\color {Chapter }Opposite Quasigroups and Loops}}{21}{section.4.12}
\contentsline {subsection}{\numberline {4.12.1}\leavevmode {\color {Chapter }Opposite, OppositeQuasigroup and OppositeLoop}}{21}{subsection.4.12.1}
\contentsline {chapter}{\numberline {5}\leavevmode {\color {Chapter }Basic Methods And Attributes}}{22}{chapter.5}
\contentsline {section}{\numberline {5.1}\leavevmode {\color {Chapter }Basic Attributes}}{22}{section.5.1}
\contentsline {subsection}{\numberline {5.1.1}\leavevmode {\color {Chapter }Elements}}{22}{subsection.5.1.1}
\contentsline {subsection}{\numberline {5.1.2}\leavevmode {\color {Chapter }CayleyTable}}{22}{subsection.5.1.2}
\contentsline {subsection}{\numberline {5.1.3}\leavevmode {\color {Chapter }One}}{22}{subsection.5.1.3}
\contentsline {subsection}{\numberline {5.1.4}\leavevmode {\color {Chapter }Size}}{22}{subsection.5.1.4}
\contentsline {subsection}{\numberline {5.1.5}\leavevmode {\color {Chapter }Exponent}}{23}{subsection.5.1.5}
\contentsline {section}{\numberline {5.2}\leavevmode {\color {Chapter }Basic Arithmetic Operations}}{23}{section.5.2}
\contentsline {subsection}{\numberline {5.2.1}\leavevmode {\color {Chapter }LeftDivision and RightDivision}}{23}{subsection.5.2.1}
\contentsline {subsection}{\numberline {5.2.2}\leavevmode {\color {Chapter }LeftDivisionCayleyTable and RightDivisionCayleyTable}}{23}{subsection.5.2.2}
\contentsline {section}{\numberline {5.3}\leavevmode {\color {Chapter }Powers and Inverses}}{23}{section.5.3}
\contentsline {subsection}{\numberline {5.3.1}\leavevmode {\color {Chapter }LeftInverse, RightInverse and Inverse}}{24}{subsection.5.3.1}
\contentsline {section}{\numberline {5.4}\leavevmode {\color {Chapter }Associators and Commutators}}{24}{section.5.4}
\contentsline {subsection}{\numberline {5.4.1}\leavevmode {\color {Chapter }Associator}}{24}{subsection.5.4.1}
\contentsline {subsection}{\numberline {5.4.2}\leavevmode {\color {Chapter }Commutator}}{24}{subsection.5.4.2}
\contentsline {section}{\numberline {5.5}\leavevmode {\color {Chapter }Generators}}{24}{section.5.5}
\contentsline {subsection}{\numberline {5.5.1}\leavevmode {\color {Chapter }GeneratorsOfQuasigroup and GeneratorsOfLoop}}{24}{subsection.5.5.1}
\contentsline {subsection}{\numberline {5.5.2}\leavevmode {\color {Chapter }GeneratorsSmallest}}{25}{subsection.5.5.2}
\contentsline {subsection}{\numberline {5.5.3}\leavevmode {\color {Chapter }SmallGeneratingSet}}{25}{subsection.5.5.3}
\contentsline {chapter}{\numberline {6}\leavevmode {\color {Chapter }Methods Based on Permutation Groups}}{26}{chapter.6}
\contentsline {section}{\numberline {6.1}\leavevmode {\color {Chapter }Parent of a Quasigroup}}{26}{section.6.1}
\contentsline {subsection}{\numberline {6.1.1}\leavevmode {\color {Chapter }Parent}}{26}{subsection.6.1.1}
\contentsline {subsection}{\numberline {6.1.2}\leavevmode {\color {Chapter }Position}}{26}{subsection.6.1.2}
\contentsline {subsection}{\numberline {6.1.3}\leavevmode {\color {Chapter }PosInParent}}{27}{subsection.6.1.3}
\contentsline {section}{\numberline {6.2}\leavevmode {\color {Chapter }Subquasigroups and Subloops}}{27}{section.6.2}
\contentsline {subsection}{\numberline {6.2.1}\leavevmode {\color {Chapter }Subquasigroup}}{27}{subsection.6.2.1}
\contentsline {subsection}{\numberline {6.2.2}\leavevmode {\color {Chapter }Subloop}}{27}{subsection.6.2.2}
\contentsline {subsection}{\numberline {6.2.3}\leavevmode {\color {Chapter }IsSubquasigroup and IsSubloop}}{27}{subsection.6.2.3}
\contentsline {subsection}{\numberline {6.2.4}\leavevmode {\color {Chapter }AllSubquasigroups}}{27}{subsection.6.2.4}
\contentsline {subsection}{\numberline {6.2.5}\leavevmode {\color {Chapter }AllSubloops}}{28}{subsection.6.2.5}
\contentsline {subsection}{\numberline {6.2.6}\leavevmode {\color {Chapter }RightCosets}}{28}{subsection.6.2.6}
\contentsline {subsection}{\numberline {6.2.7}\leavevmode {\color {Chapter }RightTransversal}}{28}{subsection.6.2.7}
\contentsline {section}{\numberline {6.3}\leavevmode {\color {Chapter }Translations and Sections}}{28}{section.6.3}
\contentsline {subsection}{\numberline {6.3.1}\leavevmode {\color {Chapter }LeftTranslation and RightTranslation}}{28}{subsection.6.3.1}
\contentsline {subsection}{\numberline {6.3.2}\leavevmode {\color {Chapter }LeftSection and RightSection}}{28}{subsection.6.3.2}
\contentsline {section}{\numberline {6.4}\leavevmode {\color {Chapter }Multiplication Groups}}{29}{section.6.4}
\contentsline {subsection}{\numberline {6.4.1}\leavevmode {\color {Chapter }LeftMutliplicationGroup, RightMultiplicationGroup and MultiplicationGroup}}{29}{subsection.6.4.1}
\contentsline {subsection}{\numberline {6.4.2}\leavevmode {\color {Chapter }RelativeLeftMultiplicationGroup, RelativeRightMultiplicationGroup and RelativeMultiplicationGroup}}{29}{subsection.6.4.2}
\contentsline {section}{\numberline {6.5}\leavevmode {\color {Chapter }Inner Mapping Groups}}{29}{section.6.5}
\contentsline {subsection}{\numberline {6.5.1}\leavevmode {\color {Chapter }LeftInnerMapping, RightInnerMapping, MiddleInnerMapping}}{30}{subsection.6.5.1}
\contentsline {subsection}{\numberline {6.5.2}\leavevmode {\color {Chapter }LeftInnerMappingGroup, RightInnerMappingGroup, MiddleInnerMappingGroup}}{30}{subsection.6.5.2}
\contentsline {subsection}{\numberline {6.5.3}\leavevmode {\color {Chapter }InnerMappingGroup}}{30}{subsection.6.5.3}
\contentsline {section}{\numberline {6.6}\leavevmode {\color {Chapter }Nuclei, Commutant, Center, and Associator Subloop}}{30}{section.6.6}
\contentsline {subsection}{\numberline {6.6.1}\leavevmode {\color {Chapter }LeftNucles, MiddleNucleus, and RightNucleus}}{30}{subsection.6.6.1}
\contentsline {subsection}{\numberline {6.6.2}\leavevmode {\color {Chapter }Nuc, NucleusOfQuasigroup and NucleusOfLoop}}{31}{subsection.6.6.2}
\contentsline {subsection}{\numberline {6.6.3}\leavevmode {\color {Chapter }Commutant}}{31}{subsection.6.6.3}
\contentsline {subsection}{\numberline {6.6.4}\leavevmode {\color {Chapter }Center}}{31}{subsection.6.6.4}
\contentsline {subsection}{\numberline {6.6.5}\leavevmode {\color {Chapter }AssociatorSubloop}}{31}{subsection.6.6.5}
\contentsline {section}{\numberline {6.7}\leavevmode {\color {Chapter }Normal Subloops and Simple Loops}}{31}{section.6.7}
\contentsline {subsection}{\numberline {6.7.1}\leavevmode {\color {Chapter }IsNormal}}{31}{subsection.6.7.1}
\contentsline {subsection}{\numberline {6.7.2}\leavevmode {\color {Chapter }NormalClosure}}{31}{subsection.6.7.2}
\contentsline {subsection}{\numberline {6.7.3}\leavevmode {\color {Chapter }IsSimple}}{32}{subsection.6.7.3}
\contentsline {section}{\numberline {6.8}\leavevmode {\color {Chapter }Factor Loops}}{32}{section.6.8}
\contentsline {subsection}{\numberline {6.8.1}\leavevmode {\color {Chapter }FactorLoop}}{32}{subsection.6.8.1}
\contentsline {subsection}{\numberline {6.8.2}\leavevmode {\color {Chapter }NaturalHomomorphismByNormalSubloop}}{32}{subsection.6.8.2}
\contentsline {section}{\numberline {6.9}\leavevmode {\color {Chapter }Nilpotency and Central Series}}{32}{section.6.9}
\contentsline {subsection}{\numberline {6.9.1}\leavevmode {\color {Chapter }IsNilpotent}}{32}{subsection.6.9.1}
\contentsline {subsection}{\numberline {6.9.2}\leavevmode {\color {Chapter }NilpotencyClassOfLoop}}{32}{subsection.6.9.2}
\contentsline {subsection}{\numberline {6.9.3}\leavevmode {\color {Chapter }IsStronglyNilpotent}}{32}{subsection.6.9.3}
\contentsline {subsection}{\numberline {6.9.4}\leavevmode {\color {Chapter }UpperCentralSeries}}{33}{subsection.6.9.4}
\contentsline {subsection}{\numberline {6.9.5}\leavevmode {\color {Chapter }LowerCentralSeries}}{33}{subsection.6.9.5}
\contentsline {section}{\numberline {6.10}\leavevmode {\color {Chapter }Solvability, Derived Series and Frattini Subloop}}{33}{section.6.10}
\contentsline {subsection}{\numberline {6.10.1}\leavevmode {\color {Chapter }IsSolvable}}{33}{subsection.6.10.1}
\contentsline {subsection}{\numberline {6.10.2}\leavevmode {\color {Chapter }DerivedSubloop}}{33}{subsection.6.10.2}
\contentsline {subsection}{\numberline {6.10.3}\leavevmode {\color {Chapter }DerivedLength}}{33}{subsection.6.10.3}
\contentsline {subsection}{\numberline {6.10.4}\leavevmode {\color {Chapter }FrattiniSubloop and FrattinifactorSize}}{33}{subsection.6.10.4}
\contentsline {subsection}{\numberline {6.10.5}\leavevmode {\color {Chapter }FrattinifactorSize}}{33}{subsection.6.10.5}
\contentsline {section}{\numberline {6.11}\leavevmode {\color {Chapter }Isomorphisms and Automorphisms}}{33}{section.6.11}
\contentsline {subsection}{\numberline {6.11.1}\leavevmode {\color {Chapter }IsomorphismQuasigroups}}{33}{subsection.6.11.1}
\contentsline {subsection}{\numberline {6.11.2}\leavevmode {\color {Chapter }IsomorphismLoops}}{34}{subsection.6.11.2}
\contentsline {subsection}{\numberline {6.11.3}\leavevmode {\color {Chapter }QuasigroupsUpToIsomorphism}}{34}{subsection.6.11.3}
\contentsline {subsection}{\numberline {6.11.4}\leavevmode {\color {Chapter }LoopsUpToIsomorphism}}{34}{subsection.6.11.4}
\contentsline {subsection}{\numberline {6.11.5}\leavevmode {\color {Chapter }AutomorphismGroup}}{34}{subsection.6.11.5}
\contentsline {subsection}{\numberline {6.11.6}\leavevmode {\color {Chapter }IsomorphicCopyByPerm}}{34}{subsection.6.11.6}
\contentsline {subsection}{\numberline {6.11.7}\leavevmode {\color {Chapter }IsomorphicCopyByNormalSubloop}}{34}{subsection.6.11.7}
\contentsline {subsection}{\numberline {6.11.8}\leavevmode {\color {Chapter }Discriminator}}{35}{subsection.6.11.8}
\contentsline {subsection}{\numberline {6.11.9}\leavevmode {\color {Chapter }AreEqualDiscriminators}}{35}{subsection.6.11.9}
\contentsline {section}{\numberline {6.12}\leavevmode {\color {Chapter }Isotopisms}}{35}{section.6.12}
\contentsline {subsection}{\numberline {6.12.1}\leavevmode {\color {Chapter }IsotopismLoops}}{35}{subsection.6.12.1}
\contentsline {subsection}{\numberline {6.12.2}\leavevmode {\color {Chapter }LoopsUpToIsotopism}}{35}{subsection.6.12.2}
\contentsline {chapter}{\numberline {7}\leavevmode {\color {Chapter }Testing Properties of Quasigroups and Loops}}{36}{chapter.7}
\contentsline {section}{\numberline {7.1}\leavevmode {\color {Chapter }Associativity, Commutativity and Generalizations}}{36}{section.7.1}
\contentsline {subsection}{\numberline {7.1.1}\leavevmode {\color {Chapter }IsAssociative}}{36}{subsection.7.1.1}
\contentsline {subsection}{\numberline {7.1.2}\leavevmode {\color {Chapter }IsCommutative}}{36}{subsection.7.1.2}
\contentsline {subsection}{\numberline {7.1.3}\leavevmode {\color {Chapter }IsPowerAssociative}}{36}{subsection.7.1.3}
\contentsline {subsection}{\numberline {7.1.4}\leavevmode {\color {Chapter }IsDiassociative}}{36}{subsection.7.1.4}
\contentsline {section}{\numberline {7.2}\leavevmode {\color {Chapter }Inverse Propeties}}{37}{section.7.2}
\contentsline {subsection}{\numberline {7.2.1}\leavevmode {\color {Chapter }HasLeftInverseProperty, HasRightInverseProperty and HasInverseProperty}}{37}{subsection.7.2.1}
\contentsline {subsection}{\numberline {7.2.2}\leavevmode {\color {Chapter }HasTwosidedInverses}}{37}{subsection.7.2.2}
\contentsline {subsection}{\numberline {7.2.3}\leavevmode {\color {Chapter }HasWeakInverseProperty}}{37}{subsection.7.2.3}
\contentsline {subsection}{\numberline {7.2.4}\leavevmode {\color {Chapter }HasAutomorphicInverseProperty}}{37}{subsection.7.2.4}
\contentsline {subsection}{\numberline {7.2.5}\leavevmode {\color {Chapter }HasAntiautomorphicInverseProperty}}{37}{subsection.7.2.5}
\contentsline {section}{\numberline {7.3}\leavevmode {\color {Chapter }Some Properties of Quasigroups}}{38}{section.7.3}
\contentsline {subsection}{\numberline {7.3.1}\leavevmode {\color {Chapter }IsSemisymmetric}}{38}{subsection.7.3.1}
\contentsline {subsection}{\numberline {7.3.2}\leavevmode {\color {Chapter }IsTotallySymmetric}}{38}{subsection.7.3.2}
\contentsline {subsection}{\numberline {7.3.3}\leavevmode {\color {Chapter }IsIdempotent}}{38}{subsection.7.3.3}
\contentsline {subsection}{\numberline {7.3.4}\leavevmode {\color {Chapter }IsSteinerQuasigroup}}{38}{subsection.7.3.4}
\contentsline {subsection}{\numberline {7.3.5}\leavevmode {\color {Chapter }IsUnipotent}}{38}{subsection.7.3.5}
\contentsline {subsection}{\numberline {7.3.6}\leavevmode {\color {Chapter }IsLeftDistributive, IsRightDistributive, IsDistributive}}{38}{subsection.7.3.6}
\contentsline {subsection}{\numberline {7.3.7}\leavevmode {\color {Chapter }IsEntropic and IsMedial}}{39}{subsection.7.3.7}
\contentsline {section}{\numberline {7.4}\leavevmode {\color {Chapter }Loops of Bol Moufang Type}}{39}{section.7.4}
\contentsline {subsection}{\numberline {7.4.1}\leavevmode {\color {Chapter }IsExtraLoop}}{40}{subsection.7.4.1}
\contentsline {subsection}{\numberline {7.4.2}\leavevmode {\color {Chapter }IsMoufangLoop}}{40}{subsection.7.4.2}
\contentsline {subsection}{\numberline {7.4.3}\leavevmode {\color {Chapter }IsCLoop}}{40}{subsection.7.4.3}
\contentsline {subsection}{\numberline {7.4.4}\leavevmode {\color {Chapter }IsLeftBolLoop}}{40}{subsection.7.4.4}
\contentsline {subsection}{\numberline {7.4.5}\leavevmode {\color {Chapter }IsRightBolLoop}}{40}{subsection.7.4.5}
\contentsline {subsection}{\numberline {7.4.6}\leavevmode {\color {Chapter }IsLCLoop}}{40}{subsection.7.4.6}
\contentsline {subsection}{\numberline {7.4.7}\leavevmode {\color {Chapter }IsRCLoop}}{40}{subsection.7.4.7}
\contentsline {subsection}{\numberline {7.4.8}\leavevmode {\color {Chapter }IsLeftNuclearSquareLoop}}{40}{subsection.7.4.8}
\contentsline {subsection}{\numberline {7.4.9}\leavevmode {\color {Chapter }IsMiddleNuclearSquareLoop}}{40}{subsection.7.4.9}
\contentsline {subsection}{\numberline {7.4.10}\leavevmode {\color {Chapter }IsRightNuclearSquareLoop}}{40}{subsection.7.4.10}
\contentsline {subsection}{\numberline {7.4.11}\leavevmode {\color {Chapter }IsNuclearSquareLoop}}{41}{subsection.7.4.11}
\contentsline {subsection}{\numberline {7.4.12}\leavevmode {\color {Chapter }IsFlexible}}{41}{subsection.7.4.12}
\contentsline {subsection}{\numberline {7.4.13}\leavevmode {\color {Chapter }IsLeftAlternative}}{41}{subsection.7.4.13}
\contentsline {subsection}{\numberline {7.4.14}\leavevmode {\color {Chapter }IsRightAlternative}}{41}{subsection.7.4.14}
\contentsline {subsection}{\numberline {7.4.15}\leavevmode {\color {Chapter }IsAlternative}}{41}{subsection.7.4.15}
\contentsline {section}{\numberline {7.5}\leavevmode {\color {Chapter }Power Alternative Loops}}{42}{section.7.5}
\contentsline {subsection}{\numberline {7.5.1}\leavevmode {\color {Chapter }IsLeftPowerAlternative, IsRightPowerAlternative and IsPowerAlternative}}{42}{subsection.7.5.1}
\contentsline {section}{\numberline {7.6}\leavevmode {\color {Chapter }Conjugacy Closed Loops and Related Properties}}{42}{section.7.6}
\contentsline {subsection}{\numberline {7.6.1}\leavevmode {\color {Chapter }IsLCCLoop}}{42}{subsection.7.6.1}
\contentsline {subsection}{\numberline {7.6.2}\leavevmode {\color {Chapter }IsRCCLoop}}{42}{subsection.7.6.2}
\contentsline {subsection}{\numberline {7.6.3}\leavevmode {\color {Chapter }IsCCLoop}}{42}{subsection.7.6.3}
\contentsline {subsection}{\numberline {7.6.4}\leavevmode {\color {Chapter }IsOsbornLoop}}{42}{subsection.7.6.4}
\contentsline {section}{\numberline {7.7}\leavevmode {\color {Chapter }Automorphic Loops}}{43}{section.7.7}
\contentsline {subsection}{\numberline {7.7.1}\leavevmode {\color {Chapter }IsLeftAutomorphicLoop}}{43}{subsection.7.7.1}
\contentsline {subsection}{\numberline {7.7.2}\leavevmode {\color {Chapter }IsMiddleAutomorphicLoop}}{43}{subsection.7.7.2}
\contentsline {subsection}{\numberline {7.7.3}\leavevmode {\color {Chapter }IsRightAutomorphicLoop}}{44}{subsection.7.7.3}
\contentsline {subsection}{\numberline {7.7.4}\leavevmode {\color {Chapter }IsAutomorphicLoop}}{44}{subsection.7.7.4}
\contentsline {section}{\numberline {7.8}\leavevmode {\color {Chapter }Additonal Varieties of Loops}}{44}{section.7.8}
\contentsline {subsection}{\numberline {7.8.1}\leavevmode {\color {Chapter }IsCodeLoop}}{44}{subsection.7.8.1}
\contentsline {subsection}{\numberline {7.8.2}\leavevmode {\color {Chapter }IsSteinerLoop}}{44}{subsection.7.8.2}
\contentsline {subsection}{\numberline {7.8.3}\leavevmode {\color {Chapter }IsLeftBruckLoop and IsLeftKLoop}}{44}{subsection.7.8.3}
\contentsline {subsection}{\numberline {7.8.4}\leavevmode {\color {Chapter }IsRightBruckLoop and IsRightKLoop}}{44}{subsection.7.8.4}
\contentsline {chapter}{\numberline {8}\leavevmode {\color {Chapter }Specific Methods}}{45}{chapter.8}
\contentsline {section}{\numberline {8.1}\leavevmode {\color {Chapter }Core Methods for Bol Loops}}{45}{section.8.1}
\contentsline {subsection}{\numberline {8.1.1}\leavevmode {\color {Chapter }AssociatedLeftBruckLoop and AssociatedRightBruckLoop}}{45}{subsection.8.1.1}
\contentsline {subsection}{\numberline {8.1.2}\leavevmode {\color {Chapter }IsExactGroupFactorization}}{45}{subsection.8.1.2}
\contentsline {subsection}{\numberline {8.1.3}\leavevmode {\color {Chapter }RightBolLoopByExactGroupFactorization}}{45}{subsection.8.1.3}
\contentsline {section}{\numberline {8.2}\leavevmode {\color {Chapter }Moufang Modifications}}{46}{section.8.2}
\contentsline {subsection}{\numberline {8.2.1}\leavevmode {\color {Chapter }LoopByCyclicModification}}{46}{subsection.8.2.1}
\contentsline {subsection}{\numberline {8.2.2}\leavevmode {\color {Chapter }LoopByDihedralModification}}{46}{subsection.8.2.2}
\contentsline {subsection}{\numberline {8.2.3}\leavevmode {\color {Chapter }LoopMG2}}{46}{subsection.8.2.3}
\contentsline {section}{\numberline {8.3}\leavevmode {\color {Chapter }Triality for Moufang Loops}}{46}{section.8.3}
\contentsline {subsection}{\numberline {8.3.1}\leavevmode {\color {Chapter }TrialityPermGroup}}{47}{subsection.8.3.1}
\contentsline {subsection}{\numberline {8.3.2}\leavevmode {\color {Chapter }TrialityPcGroup}}{47}{subsection.8.3.2}
\contentsline {section}{\numberline {8.4}\leavevmode {\color {Chapter }Realizing Groups as Multiplication Groups of Loops}}{47}{section.8.4}
\contentsline {subsection}{\numberline {8.4.1}\leavevmode {\color {Chapter }AllLoopTablesInGroup}}{47}{subsection.8.4.1}
\contentsline {subsection}{\numberline {8.4.2}\leavevmode {\color {Chapter }AllProperLoopTablesInGroup}}{47}{subsection.8.4.2}
\contentsline {subsection}{\numberline {8.4.3}\leavevmode {\color {Chapter }OneLoopTableInGroup}}{47}{subsection.8.4.3}
\contentsline {subsection}{\numberline {8.4.4}\leavevmode {\color {Chapter }OneProperLoopTableInGroup}}{48}{subsection.8.4.4}
\contentsline {subsection}{\numberline {8.4.5}\leavevmode {\color {Chapter }AllLoopsWithMltGroup}}{48}{subsection.8.4.5}
\contentsline {subsection}{\numberline {8.4.6}\leavevmode {\color {Chapter }OneLoopWithMltGroup}}{48}{subsection.8.4.6}
\contentsline {chapter}{\numberline {9}\leavevmode {\color {Chapter }Libraries of Loops}}{49}{chapter.9}
\contentsline {section}{\numberline {9.1}\leavevmode {\color {Chapter }A Typical Library}}{49}{section.9.1}
\contentsline {subsection}{\numberline {9.1.1}\leavevmode {\color {Chapter }LibraryLoop}}{49}{subsection.9.1.1}
\contentsline {subsection}{\numberline {9.1.2}\leavevmode {\color {Chapter }MyLibraryLoop}}{49}{subsection.9.1.2}
\contentsline {subsection}{\numberline {9.1.3}\leavevmode {\color {Chapter }DisplayLibraryInfo}}{50}{subsection.9.1.3}
\contentsline {section}{\numberline {9.2}\leavevmode {\color {Chapter }Left Bol Loops and Right Bol Loops}}{50}{section.9.2}
\contentsline {subsection}{\numberline {9.2.1}\leavevmode {\color {Chapter }LeftBolLoop}}{50}{subsection.9.2.1}
\contentsline {subsection}{\numberline {9.2.2}\leavevmode {\color {Chapter }RightBolLoop}}{50}{subsection.9.2.2}
\contentsline {section}{\numberline {9.3}\leavevmode {\color {Chapter }Moufang Loops}}{50}{section.9.3}
\contentsline {subsection}{\numberline {9.3.1}\leavevmode {\color {Chapter }MoufangLoop}}{50}{subsection.9.3.1}
\contentsline {section}{\numberline {9.4}\leavevmode {\color {Chapter }Code Loops}}{51}{section.9.4}
\contentsline {subsection}{\numberline {9.4.1}\leavevmode {\color {Chapter }CodeLoop}}{51}{subsection.9.4.1}
\contentsline {section}{\numberline {9.5}\leavevmode {\color {Chapter }Steiner Loops}}{51}{section.9.5}
\contentsline {subsection}{\numberline {9.5.1}\leavevmode {\color {Chapter }SteinerLoop}}{51}{subsection.9.5.1}
\contentsline {section}{\numberline {9.6}\leavevmode {\color {Chapter }Conjugacy Closed Loops}}{51}{section.9.6}
\contentsline {subsection}{\numberline {9.6.1}\leavevmode {\color {Chapter }RCCLoop and RightConjugacyClosedLoop}}{52}{subsection.9.6.1}
\contentsline {subsection}{\numberline {9.6.2}\leavevmode {\color {Chapter }LCCLoop and LeftConjugacyClosedLoop}}{52}{subsection.9.6.2}
\contentsline {subsection}{\numberline {9.6.3}\leavevmode {\color {Chapter }CCLoop and ConjugacyClosedLoop}}{52}{subsection.9.6.3}
\contentsline {section}{\numberline {9.7}\leavevmode {\color {Chapter }Small Loops}}{52}{section.9.7}
\contentsline {subsection}{\numberline {9.7.1}\leavevmode {\color {Chapter }SmallLoop}}{53}{subsection.9.7.1}
\contentsline {section}{\numberline {9.8}\leavevmode {\color {Chapter }Paige Loops}}{53}{section.9.8}
\contentsline {subsection}{\numberline {9.8.1}\leavevmode {\color {Chapter }PaigeLoop}}{53}{subsection.9.8.1}
\contentsline {section}{\numberline {9.9}\leavevmode {\color {Chapter }Nilpotent Loops}}{53}{section.9.9}
\contentsline {subsection}{\numberline {9.9.1}\leavevmode {\color {Chapter }NilpotentLoop}}{53}{subsection.9.9.1}
\contentsline {section}{\numberline {9.10}\leavevmode {\color {Chapter }Automorphic Loops}}{53}{section.9.10}
\contentsline {subsection}{\numberline {9.10.1}\leavevmode {\color {Chapter }AutomorphicLoop}}{53}{subsection.9.10.1}
\contentsline {section}{\numberline {9.11}\leavevmode {\color {Chapter }Interesting Loops}}{54}{section.9.11}
\contentsline {subsection}{\numberline {9.11.1}\leavevmode {\color {Chapter }InterestingLoop}}{54}{subsection.9.11.1}
\contentsline {section}{\numberline {9.12}\leavevmode {\color {Chapter }Libraries of Loops Up To Isotopism}}{54}{section.9.12}
\contentsline {subsection}{\numberline {9.12.1}\leavevmode {\color {Chapter }ItpSmallLoop}}{54}{subsection.9.12.1}
\contentsline {chapter}{\numberline {A}\leavevmode {\color {Chapter }Files}}{55}{appendix.A}
\contentsline {chapter}{\numberline {B}\leavevmode {\color {Chapter }Filters}}{57}{appendix.B}
\contentsline {chapter}{References}{61}{appendix*.3}
\contentsline {chapter}{Index}{62}{section*.4}

62
doc/loops.xml
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@ -6,7 +6,8 @@
<!-- Read the file pkg/loops/etc/gapdoc.txt for instructions on how to produce the documentation. --> <!-- Read the file pkg/loops/etc/gapdoc.txt for instructions on how to produce the documentation. -->
<!-- Typesetting rules for this document that produce acceptable outcome in both html and pdf: <!-- Typesetting rules for this document that produce acceptable outcome in
both html and pdf:
* Use <Br/><P/> at the beginning of paragraphs that are supposed to have an empty line just before them for greater emphasis, eg., before Remark, Example, and to break up the text. (This has no effect on html.) * Use <Br/><P/> at the beginning of paragraphs that are supposed to have an empty line just before them for greater emphasis, eg., before Remark, Example, and to break up the text. (This has no effect on html.)
@ -24,7 +25,7 @@
<TitlePage> <TitlePage>
<Title>The <Package>LOOPS</Package> Package</Title> <Title>The <Package>LOOPS</Package> Package</Title>
<Version>Version 3.3.0</Version> <Version>Version 3.4.0</Version>
<Subtitle>Computing with quasigroups and loops in &GAP;</Subtitle> <Subtitle>Computing with quasigroups and loops in &GAP;</Subtitle>
<Author>G&#225;bor P. Nagy <Author>G&#225;bor P. Nagy
<Email>nagyg@math.u-szeged.hu</Email> <Email>nagyg@math.u-szeged.hu</Email>
@ -34,7 +35,7 @@
<Email>petr@math.du.edu</Email> <Email>petr@math.du.edu</Email>
<Address>Department of Mathematics, University of Denver</Address> <Address>Department of Mathematics, University of Denver</Address>
</Author> </Author>
<Copyright>&copyright; 2016 G&#225;bor P. Nagy and Petr Vojt&#283;chovsk&#253;. <Copyright>&copyright; 2017 G&#225;bor P. Nagy and Petr Vojt&#283;chovsk&#253;.
</Copyright> </Copyright>
</TitlePage> </TitlePage>
@ -65,7 +66,7 @@
<Section Label="Sec:Installation"> <Heading>Installation</Heading> <Section Label="Sec:Installation"> <Heading>Installation</Heading>
Have <Package>GAP 4.7</Package> or newer installed on your computer. Have <Package>GAP 4.8</Package> or newer installed on your computer.
<P/>If you do not see the subfolder <File>pkg/loops</File> in the main directory of &GAP; then download the <Package>LOOPS</Package> package from the distribution website <URL>http://www.math.du.edu/loops</URL> and unpack the downloaded file into the <File>pkg</File> subfolder. <P/>If you do not see the subfolder <File>pkg/loops</File> in the main directory of &GAP; then download the <Package>LOOPS</Package> package from the distribution website <URL>http://www.math.du.edu/loops</URL> and unpack the downloaded file into the <File>pkg</File> subfolder.
@ -124,9 +125,9 @@ We welcome all comments and suggestions on <Package>LOOPS</Package>, especially
<Section Label="Sec:Acknowledgment"> <Heading>Acknowledgment</Heading> <Section Label="Sec:Acknowledgment"> <Heading>Acknowledgment</Heading>
We thank the following people for sending us remarks and comments, and for suggesting new functionality of the package: Muniru Asiru, Bjoern Assmann, Andreas Distler, Ale&#353; Dr&#225;pal, Steve Flammia, Kenneth W. Johnson, Michael K. Kinyon, Alexander Konovalov, Frank L&#252;beck and Jonathan D.H. Smith. We thank the following people for sending us remarks and comments, and for suggesting new functionality of the package: Muniru Asiru, Bjoern Assmann, Andreas Distler, Ale&#353; Dr&#225;pal, Graham Ellis, Steve Flammia, Kenneth W. Johnson, Michael K. Kinyon, Alexander Konovalov, Frank L&#252;beck, Jonathan D.H. Smith, David Stanovsk&#253; and Glen Whitney.
<P/>The library of Moufang loops of order 243 was generated from data provided by Michael C. Slattery and Ashley L. Zenisek. The library of right conjugacy closed loops of order less than 28 was generated from data provided by Katharina Artic. The library of commutative automorphic loops of order 27, 81 and 243 was obtained jointly with Izabella Stuhl. <P/>The library of Moufang loops of order 243 was generated from data provided by Michael C. Slattery and Ashley L. Zenisek. The library of right conjugacy closed loops of order less than 28 was generated from data provided by Katharina Artic. The library of right Bruck loops of order 27, 81 was obtained jointly with Izabella Stuhl.
<P/>G&#225;bor P. Nagy was supported by OTKA grants F042959 and T043758, and Petr Vojt&#283;chovsk&#253; was supported by the 2006 and 2016 University of Denver PROF grants and the Simons Foundation Collaboration Grant 210176. <P/>G&#225;bor P. Nagy was supported by OTKA grants F042959 and T043758, and Petr Vojt&#283;chovsk&#253; was supported by the 2006 and 2016 University of Denver PROF grants and the Simons Foundation Collaboration Grant 210176.
@ -260,8 +261,8 @@ DeclareCategory( "IsLoopElement",
DeclareRepresentation( "IsLoopElmRep", DeclareRepresentation( "IsLoopElmRep",
IsPositionalObjectRep and IsMultiplicativeElementWithInverse, [1] ); IsPositionalObjectRep and IsMultiplicativeElementWithInverse, [1] );
## latin (auxiliary category for GAP to tell apart IsMagma and IsQuasigroup) ## latin (auxiliary category for GAP to tell apart IsMagma and IsQuasigroup)
DeclareCategory( "IsLatin", IsObject ); DeclareCategory( "IsLatinMagma", IsObject );
DeclareCategory( "IsQuasigroup", IsMagma and IsLatin ); DeclareCategory( "IsQuasigroup", IsMagma and IsLatinMagma );
DeclareCategory( "IsLoop", IsQuasigroup and DeclareCategory( "IsLoop", IsQuasigroup and
IsMultiplicativeElementWithInverseCollection); IsMultiplicativeElementWithInverseCollection);
</Verb> </Verb>
@ -1325,9 +1326,19 @@ See Section <Ref Sect="Sec:NilpotenceAndSolvability"/> for definitions of solvab
<Br/><P/>While dealing with Cayley tables, it is often useful to rename or reorder the elements of the underlying quasigroup without changing the isomorphism type of the quasigroups. <Package>LOOPS</Package> contains several functions for this purpose. <Br/><P/>While dealing with Cayley tables, it is often useful to rename or reorder the elements of the underlying quasigroup without changing the isomorphism type of the quasigroups. <Package>LOOPS</Package> contains several functions for this purpose.
<ManSection>
<Oper Name="QuasigroupIsomorph" Arg="Q, f"/>
<Returns>When <Arg>Q</Arg> is a quasigroup and <Arg>f</Arg> is a permutation of <M>1,\dots,|</M><Arg>Q</Arg><M>|</M>, returns the quasigroup defined on the same set as <Arg>Q</Arg> with multiplication <M>*</M> defined by <M>x*y = </M><Arg>f</Arg><M>(</M><Arg>f</Arg><M>{}^{-1}(x)</M><Arg>f</Arg><M>{}^{-1}(y))</M>.</Returns>
</ManSection>
<ManSection>
<Oper Name="LoopIsomorph" Arg="Q, f"/>
<Returns>When <Arg>Q</Arg> is a loop and <Arg>f</Arg> is a permutation of <M>1,\dots,|</M><Arg>Q</Arg><M>|</M> fixing <M>1</M>, returns the loop defined on the same set as <Arg>Q</Arg> with multiplication <M>*</M> defined by <M>x*y = </M><Arg>f</Arg><M>(</M><Arg>f</Arg><M>{}^{-1}(x)</M><Arg>f</Arg><M>{}^{-1}(y))</M>. If <Arg>f</Arg><M>(1)=c\ne 1</M>, the isomorphism <M>(1,c)</M> is applied after <Arg>f</Arg>.</Returns>
</ManSection>
<ManSection> <ManSection>
<Oper Name="IsomorphicCopyByPerm" Arg="Q, f"/> <Oper Name="IsomorphicCopyByPerm" Arg="Q, f"/>
<Returns>When <Arg>Q</Arg> is a quasigroup and <Arg>f</Arg> is a permutation of <M>1,\dots,|</M><Arg>Q</Arg><M>|</M>, returns a quasigroup defined on the same set as <Arg>Q</Arg> with multiplication <M>*</M> defined by <M>x*y = </M><Arg>f</Arg><M>(</M><Arg>f</Arg><M>{}^{-1}(x)</M><Arg>f</Arg><M>{}^{-1}(y))</M>. When <Arg>Q</Arg> is a declared loop, a loop is returned. Consequently, when <Arg>Q</Arg> is a declared loop and <Arg>f</Arg><M>(1) = k\ne 1</M>, then <Arg>f</Arg> is first replaced with <Arg>f</Arg><M>\circ (1,k)</M>, to make sure that the resulting Cayley table is normalized.</Returns> <Returns><Code>LoopIsomorphism(<Arg>Q</Arg>,<Arg>f</Arg>)</Code> if <Arg>Q</Arg> is a loop, and <Code>QuasigroupIsomorphism(<Arg>Q</Arg>,<Arg>f</Arg>)</Code> if <Arg>Q</Arg> is a quasigroup.</Returns>
</ManSection> </ManSection>
<ManSection> <ManSection>
@ -1971,6 +1982,26 @@ The library named <Emph>left Bol</Emph> contains all nonassociative left Bol loo
</Section> </Section>
<!-- Section: Left Bruck Loops and Right Bruck Loops --------------------------------------------------- -->
<Section Label="Sec:BruckLoops"> <Heading>Left Bruck Loops and Right Bruck Loops</Heading>
The emmerging library named <Emph>left Bruck</Emph> contains all left Bruck loops of orders <M>3</M>, <M>9</M>, <M>27</M> and <M>81</M> (there are <M>1</M>, <M>2</M>, <M>7</M> and <M>72</M> such loops, respectively).
<P/>For an odd prime <M>p</M>, left Bruck loops of order <M>p^k</M> are centrally nilpotent and hence central extensions of the cyclic group of order <M>p</M> by a left Bruck loop of order <M>p^{k-1}</M>. It is known that left Bruck loops of order <M>p</M> and <M>p^2</M> are abelian groups; we have included them in the library because of the iterative nature of the construction of nilpotent loops.
<ManSection>
<Func Name="LeftBruckLoop" Arg="n, m"/>
<Returns>The <Arg>m</Arg>th left Bruck loop of order <Arg>n</Arg> in the library.</Returns>
</ManSection>
<ManSection>
<Func Name="RightBruckLoop" Arg="n, m"/>
<Returns>The <Arg>m</Arg>th right Bruck loop of order <Arg>n</Arg> in the library.</Returns>
</ManSection>
</Section>
<!-- Section: Moufang Loops ---------------------------------------------------------------------------- --> <!-- Section: Moufang Loops ---------------------------------------------------------------------------- -->
<Section Label="Sec:MoufangLoops"> <Heading>Moufang Loops</Heading> <Section Label="Sec:MoufangLoops"> <Heading>Moufang Loops</Heading>
@ -2072,7 +2103,7 @@ The following table summarizes the number of right conjugacy closed loops of a g
<Description><B>Remark:</B> Only the right conjugacy closed loops are stored in the library. Left conjugacy closed loops are obtained from right conjugacy closed loops via <Code>Opposite</Code>.<Br/></Description> <Description><B>Remark:</B> Only the right conjugacy closed loops are stored in the library. Left conjugacy closed loops are obtained from right conjugacy closed loops via <Code>Opposite</Code>.<Br/></Description>
</ManSection> </ManSection>
<P/>The library named <Emph>CC</Emph> contains all nonassociative conjugacy closed loops of order <M>n\le 27</M> and also of orders <M>2p</M> and <M>p^2</M> for all primes <M>p</M>. <P/>The library named <Emph>CC</Emph> contains all CC loops of order <M>2\le 2^k\le 64</M>, <M>3\le 3^k\le 81</M>, <M>5\le 5^k\le 125</M>, <M>7\le 7^k\le 343</M>, all nonassociative CC loops of order less than 28, and all nonassociative CC loops of order <M>p^2</M> and <M>2p</M> for any odd prime <M>p</M>.
<P/>By results of Kunen <Cite Key="Kun"/>, for every odd prime <M>p</M> there are precisely 3 nonassociative conjugacy closed loops of order <M>p^2</M>. Cs&#246;rg&#337; and Dr&#225;pal <Cite Key="CsDr"/> described these 3 loops by multiplicative formulas on <M>\mathbb{Z}_{p^2}</M> and <M>\mathbb{Z}_p \times \mathbb{Z}_p</M> as follows: <P/>By results of Kunen <Cite Key="Kun"/>, for every odd prime <M>p</M> there are precisely 3 nonassociative conjugacy closed loops of order <M>p^2</M>. Cs&#246;rg&#337; and Dr&#225;pal <Cite Key="CsDr"/> described these 3 loops by multiplicative formulas on <M>\mathbb{Z}_{p^2}</M> and <M>\mathbb{Z}_p \times \mathbb{Z}_p</M> as follows:
<List> <List>
@ -2141,7 +2172,9 @@ are 2623755 nilpotent loops of order 12, and 123794003928541545927226368 nilpote
<Section Label="Sec:AutomorphicLoops"> <Heading>Automorphic Loops</Heading> <Section Label="Sec:AutomorphicLoops"> <Heading>Automorphic Loops</Heading>
The library named <Emph>automorphic</Emph> contains all nonassociative automorphic loops of order less than 16 up to isomorphism (there is 1 such loop of order 6, 7 of order 8, 3 of order 10, 2 of order 12, 5 of order 14, and 2 of order 15), all commutative automorphic loops of order 3, 9, 27 and 81 (there are 1, 2, 7 and 72 such loops, respectively, including abelian groups), and commutative automorphic loops <M>Q</M> of order 243 possessing a central subloop <M>S</M> of order 3 such that <M>Q/S</M> is not the elementary abelian group of order 81 (there are 118451 such loops). The library named <Emph>automorphic</Emph> contains all nonassociative automorphic loops of order less than 16 up to isomorphism (there is 1 such loop of order 6, 7 of order 8, 3 of order 10, 2 of order 12, 5 of order 14, and 2 of order 15) and all commutative automorphic loops of order 3, 9, 27 and 81 (there are 1, 2, 7 and 72 such loops).
<P/>It turns out that commutative automorphic loops of order 3, 9, 27 and 81 (but not 243) are in one-to-on correspondence with left Bruck loops of the respective orders, see <Cite Key="Greer"/>, <Cite Key="StuhlVojtechovsky"/>. Only the left Bruck loops are stored in the library.
<ManSection> <ManSection>
<Func Name="AutomorphicLoop" Arg="n, m"/> <Func Name="AutomorphicLoop" Arg="n, m"/>
@ -2343,9 +2376,6 @@ Many implications among properties of loops are built directly into <Package>LOO
<Br/><Code>( IsLeftAutomorphicLoop, IsAutomorphicLoop )</Code> <Br/><Code>( IsLeftAutomorphicLoop, IsAutomorphicLoop )</Code>
<Br/><Code>( IsRightAutomorphicLoop, IsAutomorphicLoop )</Code> <Br/><Code>( IsRightAutomorphicLoop, IsAutomorphicLoop )</Code>
<Br/><Code>( IsMiddleAutomorphicLoop, IsAutomorphicLoop )</Code> <Br/><Code>( IsMiddleAutomorphicLoop, IsAutomorphicLoop )</Code>
<Br/><Code>( IsMiddleAutomorphicLoop, IsCommutative )</Code>
<Br/><Code>( IsAutomorphicLoop, IsLeftAutomorphicLoop and IsCommutative )</Code>
<Br/><Code>( IsAutomorphicLoop, IsRightAutomorphicLoop and IsCommutative )</Code>
<Br/><Code>( IsLeftAutomorphicLoop, IsRightAutomorphicLoop and HasAntiautomorphicInverseProperty )</Code> <Br/><Code>( IsLeftAutomorphicLoop, IsRightAutomorphicLoop and HasAntiautomorphicInverseProperty )</Code>
<Br/><Code>( IsRightAutomorphicLoop, IsLeftAutomorphicLoop and HasAntiautomorphicInverseProperty )</Code> <Br/><Code>( IsRightAutomorphicLoop, IsLeftAutomorphicLoop and HasAntiautomorphicInverseProperty )</Code>
<Br/><Code>( IsFlexible, IsMiddleAutomorphicLoop )</Code> <Br/><Code>( IsFlexible, IsMiddleAutomorphicLoop )</Code>
@ -2356,11 +2386,15 @@ Many implications among properties of loops are built directly into <Package>LOO
<Br/><Code>( IsMoufangLoop, IsAutomorphicLoop and HasLeftInverseProperty )</Code> <Br/><Code>( IsMoufangLoop, IsAutomorphicLoop and HasLeftInverseProperty )</Code>
<Br/><Code>( IsMoufangLoop, IsAutomorphicLoop and HasRightInverseProperty )</Code> <Br/><Code>( IsMoufangLoop, IsAutomorphicLoop and HasRightInverseProperty )</Code>
<Br/><Code>( IsMoufangLoop, IsAutomorphicLoop and HasWeakInverseProperty )</Code> <Br/><Code>( IsMoufangLoop, IsAutomorphicLoop and HasWeakInverseProperty )</Code>
<Br/><Code>( IsMiddleAutomorphicLoop, IsCommutative )</Code>
<Br/><Code>( IsLeftAutomorphicLoop, IsLeftBruckLoop )</Code> <Br/><Code>( IsLeftAutomorphicLoop, IsLeftBruckLoop )</Code>
<Br/><Code>( IsLeftAutomorphicLoop, IsLCCLoop )</Code> <Br/><Code>( IsLeftAutomorphicLoop, IsLCCLoop )</Code>
<Br/><Code>( IsRightAutomorphicLoop, IsRightBruckLoop )</Code> <Br/><Code>( IsRightAutomorphicLoop, IsRightBruckLoop )</Code>
<Br/><Code>( IsRightAutomorphicLoop, IsRCCLoop )</Code> <Br/><Code>( IsRightAutomorphicLoop, IsRCCLoop )</Code>
<Br/><Code>( IsAutomorphicLoop, IsCommutative and IsMoufangLoop )</Code> <Br/><Code>( IsAutomorphicLoop, IsCommutative and IsMoufangLoop )</Code>
<Br/><Code>( IsAutomorphicLoop, IsLeftAutomorphicLoop and IsMiddleAutomorphicLoop )</Code>
<Br/><Code>( IsAutomorphicLoop, IsRightAutomorphicLoop and IsMiddleAutomorphicLoop )</Code>
<Br/><Code>( IsAutomorphicLoop, IsAssociative )</Code>
</Appendix> </Appendix>

559
doc/loops_bib.xml
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@ -1,559 +0,0 @@
<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE file SYSTEM "bibxmlext.dtd">
<file>
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<author>
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</author>
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<mrreviewer>Yu. M. Movsisyan</mrreviewer>
<url>http://dx.doi.org/10.1016/j.jalgebra.2009.03.042</url>
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<other type="fjournal">Journal of Algebra</other>
</article></entry>
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@ -1,496 +0,0 @@
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[ "\033[2XIsDistributive\033[102X", "7.3-6", [ 7, 3, 6 ], 143, 38, [ "\033[2XIsDistributive\033[102X", "7.3-6", [ 7, 3, 6 ], 143, 39,
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[ "quasigroup left distributive", "7.3-6", [ 7, 3, 6 ], 143, 38, [ "quasigroup left distributive", "7.3-6", [ 7, 3, 6 ], 143, 39,
"quasigroup left distributive", "X7B76FD6E878ED4F1" ], "quasigroup left distributive", "X7B76FD6E878ED4F1" ],
[ "distributive quasigroup left", "7.3-6", [ 7, 3, 6 ], 143, 38, [ "distributive quasigroup left", "7.3-6", [ 7, 3, 6 ], 143, 39,
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[ "quasigroup right distributive", "7.3-6", [ 7, 3, 6 ], 143, 38, [ "quasigroup right distributive", "7.3-6", [ 7, 3, 6 ], 143, 39,
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[ "distributive quasigroup right", "7.3-6", [ 7, 3, 6 ], 143, 38, [ "distributive quasigroup right", "7.3-6", [ 7, 3, 6 ], 143, 39,
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[ "quasigroup distributive", "7.3-6", [ 7, 3, 6 ], 143, 38, [ "quasigroup distributive", "7.3-6", [ 7, 3, 6 ], 143, 39,
"quasigroup distributive", "X7B76FD6E878ED4F1" ], "quasigroup distributive", "X7B76FD6E878ED4F1" ],
[ "distributive quasigroup", "7.3-6", [ 7, 3, 6 ], 143, 38, [ "distributive quasigroup", "7.3-6", [ 7, 3, 6 ], 143, 39,
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[ "\033[2XIsEntropic\033[102X", "7.3-7", [ 7, 3, 7 ], 160, 39, [ "\033[2XIsEntropic\033[102X", "7.3-7", [ 7, 3, 7 ], 160, 40,
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[ "\033[2XIsMedial\033[102X", "7.3-7", [ 7, 3, 7 ], 160, 39, "ismedial", [ "\033[2XIsMedial\033[102X", "7.3-7", [ 7, 3, 7 ], 160, 40, "ismedial",
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[ "entropic quasigroup", "7.3-7", [ 7, 3, 7 ], 160, 39, [ "entropic quasigroup", "7.3-7", [ 7, 3, 7 ], 160, 40,
"entropic quasigroup", "X7F23D4D97A38D223" ], "entropic quasigroup", "X7F23D4D97A38D223" ],
[ "quasigroup entropic", "7.3-7", [ 7, 3, 7 ], 160, 39, [ "quasigroup entropic", "7.3-7", [ 7, 3, 7 ], 160, 40,
"quasigroup entropic", "X7F23D4D97A38D223" ], "quasigroup entropic", "X7F23D4D97A38D223" ],
[ "medial quasigroup", "7.3-7", [ 7, 3, 7 ], 160, 39, "medial quasigroup", [ "medial quasigroup", "7.3-7", [ 7, 3, 7 ], 160, 40, "medial quasigroup",
"X7F23D4D97A38D223" ], "X7F23D4D97A38D223" ],
[ "quasigroup medial", "7.3-7", [ 7, 3, 7 ], 160, 39, "quasigroup medial", [ "quasigroup medial", "7.3-7", [ 7, 3, 7 ], 160, 40, "quasigroup medial",
"X7F23D4D97A38D223" ], "X7F23D4D97A38D223" ],
[ "loop of Bol-Moufang type", "7.4", [ 7, 4, 0 ], 170, 39, [ "loop of Bol-Moufang type", "7.4", [ 7, 4, 0 ], 170, 40,
"loop of bol-moufang type", "X780D907986EBA6C7" ], "loop of bol-moufang type", "X780D907986EBA6C7" ],
[ "identity of Bol-Moufang type", "7.4", [ 7, 4, 0 ], 170, 39, [ "identity of Bol-Moufang type", "7.4", [ 7, 4, 0 ], 170, 40,
"identity of bol-moufang type", "X780D907986EBA6C7" ], "identity of bol-moufang type", "X780D907986EBA6C7" ],
[ "alternative loop left", "7.4", [ 7, 4, 0 ], 170, 39, [ "alternative loop left", "7.4", [ 7, 4, 0 ], 170, 40,
"alternative loop left", "X780D907986EBA6C7" ], "alternative loop left", "X780D907986EBA6C7" ],
[ "loop left alternative", "7.4", [ 7, 4, 0 ], 170, 39, [ "loop left alternative", "7.4", [ 7, 4, 0 ], 170, 40,
"loop left alternative", "X780D907986EBA6C7" ], "loop left alternative", "X780D907986EBA6C7" ],
[ "alternative loop right", "7.4", [ 7, 4, 0 ], 170, 39, [ "alternative loop right", "7.4", [ 7, 4, 0 ], 170, 40,
"alternative loop right", "X780D907986EBA6C7" ], "alternative loop right", "X780D907986EBA6C7" ],
[ "loop right alternative", "7.4", [ 7, 4, 0 ], 170, 39, [ "loop right alternative", "7.4", [ 7, 4, 0 ], 170, 40,
"loop right alternative", "X780D907986EBA6C7" ], "loop right alternative", "X780D907986EBA6C7" ],
[ "nuclear square loop left", "7.4", [ 7, 4, 0 ], 170, 39, [ "nuclear square loop left", "7.4", [ 7, 4, 0 ], 170, 40,
"nuclear square loop left", "X780D907986EBA6C7" ], "nuclear square loop left", "X780D907986EBA6C7" ],
[ "loop left nuclear square", "7.4", [ 7, 4, 0 ], 170, 39, [ "loop left nuclear square", "7.4", [ 7, 4, 0 ], 170, 40,
"loop left nuclear square", "X780D907986EBA6C7" ], "loop left nuclear square", "X780D907986EBA6C7" ],
[ "nuclear square loop middle", "7.4", [ 7, 4, 0 ], 170, 39, [ "nuclear square loop middle", "7.4", [ 7, 4, 0 ], 170, 40,
"nuclear square loop middle", "X780D907986EBA6C7" ], "nuclear square loop middle", "X780D907986EBA6C7" ],
[ "loop middle nuclear square", "7.4", [ 7, 4, 0 ], 170, 39, [ "loop middle nuclear square", "7.4", [ 7, 4, 0 ], 170, 40,
"loop middle nuclear square", "X780D907986EBA6C7" ], "loop middle nuclear square", "X780D907986EBA6C7" ],
[ "nuclear square loop right", "7.4", [ 7, 4, 0 ], 170, 39, [ "nuclear square loop right", "7.4", [ 7, 4, 0 ], 170, 40,
"nuclear square loop right", "X780D907986EBA6C7" ], "nuclear square loop right", "X780D907986EBA6C7" ],
[ "loop right nuclear square", "7.4", [ 7, 4, 0 ], 170, 39, [ "loop right nuclear square", "7.4", [ 7, 4, 0 ], 170, 40,
"loop right nuclear square", "X780D907986EBA6C7" ], "loop right nuclear square", "X780D907986EBA6C7" ],
[ "flexible loop", "7.4", [ 7, 4, 0 ], 170, 39, "flexible loop", [ "flexible loop", "7.4", [ 7, 4, 0 ], 170, 40, "flexible loop",
"X780D907986EBA6C7" ], "X780D907986EBA6C7" ],
[ "loop flexible", "7.4", [ 7, 4, 0 ], 170, 39, "loop flexible", [ "loop flexible", "7.4", [ 7, 4, 0 ], 170, 40, "loop flexible",
"X780D907986EBA6C7" ], "X780D907986EBA6C7" ],
[ "Bol loop left", "7.4", [ 7, 4, 0 ], 170, 39, "bol loop left", [ "Bol loop left", "7.4", [ 7, 4, 0 ], 170, 40, "bol loop left",
"X780D907986EBA6C7" ], "X780D907986EBA6C7" ],
[ "loop left Bol", "7.4", [ 7, 4, 0 ], 170, 39, "loop left bol", [ "loop left Bol", "7.4", [ 7, 4, 0 ], 170, 40, "loop left bol",
"X780D907986EBA6C7" ], "X780D907986EBA6C7" ],
[ "Bol loop right", "7.4", [ 7, 4, 0 ], 170, 39, "bol loop right", [ "Bol loop right", "7.4", [ 7, 4, 0 ], 170, 40, "bol loop right",
"X780D907986EBA6C7" ], "X780D907986EBA6C7" ],
[ "loop right Bol", "7.4", [ 7, 4, 0 ], 170, 39, "loop right bol", [ "loop right Bol", "7.4", [ 7, 4, 0 ], 170, 40, "loop right bol",
"X780D907986EBA6C7" ], "X780D907986EBA6C7" ],
[ "LC loop", "7.4", [ 7, 4, 0 ], 170, 39, "lc loop", "X780D907986EBA6C7" ], [ "LC loop", "7.4", [ 7, 4, 0 ], 170, 40, "lc loop", "X780D907986EBA6C7" ],
[ "loop LC", "7.4", [ 7, 4, 0 ], 170, 39, "loop lc", "X780D907986EBA6C7" ], [ "loop LC", "7.4", [ 7, 4, 0 ], 170, 40, "loop lc", "X780D907986EBA6C7" ],
[ "RC loop", "7.4", [ 7, 4, 0 ], 170, 39, "rc loop", "X780D907986EBA6C7" ], [ "RC loop", "7.4", [ 7, 4, 0 ], 170, 40, "rc loop", "X780D907986EBA6C7" ],
[ "loop RC", "7.4", [ 7, 4, 0 ], 170, 39, "loop rc", "X780D907986EBA6C7" ], [ "loop RC", "7.4", [ 7, 4, 0 ], 170, 40, "loop rc", "X780D907986EBA6C7" ],
[ "Moufang loop", "7.4", [ 7, 4, 0 ], 170, 39, "moufang loop", [ "Moufang loop", "7.4", [ 7, 4, 0 ], 170, 40, "moufang loop",
"X780D907986EBA6C7" ], "X780D907986EBA6C7" ],
[ "loop Moufang", "7.4", [ 7, 4, 0 ], 170, 39, "loop moufang", [ "loop Moufang", "7.4", [ 7, 4, 0 ], 170, 40, "loop moufang",
"X780D907986EBA6C7" ], "X780D907986EBA6C7" ],
[ "C loop", "7.4", [ 7, 4, 0 ], 170, 39, "c loop", "X780D907986EBA6C7" ], [ "C loop", "7.4", [ 7, 4, 0 ], 170, 40, "c loop", "X780D907986EBA6C7" ],
[ "loop C", "7.4", [ 7, 4, 0 ], 170, 39, "loop c", "X780D907986EBA6C7" ], [ "loop C", "7.4", [ 7, 4, 0 ], 170, 40, "loop c", "X780D907986EBA6C7" ],
[ "extra loop", "7.4", [ 7, 4, 0 ], 170, 39, "extra loop", [ "extra loop", "7.4", [ 7, 4, 0 ], 170, 40, "extra loop",
"X780D907986EBA6C7" ], "X780D907986EBA6C7" ],
[ "loop extra", "7.4", [ 7, 4, 0 ], 170, 39, "loop extra", [ "loop extra", "7.4", [ 7, 4, 0 ], 170, 40, "loop extra",
"X780D907986EBA6C7" ], "X780D907986EBA6C7" ],
[ "alternative loop", "7.4", [ 7, 4, 0 ], 170, 39, "alternative loop", [ "alternative loop", "7.4", [ 7, 4, 0 ], 170, 40, "alternative loop",
"X780D907986EBA6C7" ], "X780D907986EBA6C7" ],
[ "loop alternative", "7.4", [ 7, 4, 0 ], 170, 39, "loop alternative", [ "loop alternative", "7.4", [ 7, 4, 0 ], 170, 40, "loop alternative",
"X780D907986EBA6C7" ], "X780D907986EBA6C7" ],
[ "nuclear square loop", "7.4", [ 7, 4, 0 ], 170, 39, "nuclear square loop", [ "nuclear square loop", "7.4", [ 7, 4, 0 ], 170, 40, "nuclear square loop",
"X780D907986EBA6C7" ], "X780D907986EBA6C7" ],
[ "loop nuclear square", "7.4", [ 7, 4, 0 ], 170, 39, "loop nuclear square", [ "loop nuclear square", "7.4", [ 7, 4, 0 ], 170, 40, "loop nuclear square",
"X780D907986EBA6C7" ], "X780D907986EBA6C7" ],
[ "\033[2XIsExtraLoop\033[102X", "7.4-1", [ 7, 4, 1 ], 223, 40, [ "\033[2XIsExtraLoop\033[102X", "7.4-1", [ 7, 4, 1 ], 223, 41,
"isextraloop", "X7988AFE27D06ACB5" ], "isextraloop", "X7988AFE27D06ACB5" ],
[ "\033[2XIsMoufangLoop\033[102X", "7.4-2", [ 7, 4, 2 ], 228, 40, [ "\033[2XIsMoufangLoop\033[102X", "7.4-2", [ 7, 4, 2 ], 228, 41,
"ismoufangloop", "X7F1C151484C97E61" ], "ismoufangloop", "X7F1C151484C97E61" ],
[ "\033[2XIsCLoop\033[102X", "7.4-3", [ 7, 4, 3 ], 233, 40, "iscloop", [ "\033[2XIsCLoop\033[102X", "7.4-3", [ 7, 4, 3 ], 233, 41, "iscloop",
"X866F04DC7AE54B7C" ], "X866F04DC7AE54B7C" ],
[ "\033[2XIsLeftBolLoop\033[102X", "7.4-4", [ 7, 4, 4 ], 238, 40, [ "\033[2XIsLeftBolLoop\033[102X", "7.4-4", [ 7, 4, 4 ], 238, 41,
"isleftbolloop", "X801DAAE8834A1A65" ], "isleftbolloop", "X801DAAE8834A1A65" ],
[ "\033[2XIsRightBolLoop\033[102X", "7.4-5", [ 7, 4, 5 ], 243, 40, [ "\033[2XIsRightBolLoop\033[102X", "7.4-5", [ 7, 4, 5 ], 243, 41,
"isrightbolloop", "X79279F9787E72566" ], "isrightbolloop", "X79279F9787E72566" ],
[ "\033[2XIsLCLoop\033[102X", "7.4-6", [ 7, 4, 6 ], 248, 40, "islcloop", [ "\033[2XIsLCLoop\033[102X", "7.4-6", [ 7, 4, 6 ], 248, 41, "islcloop",
"X789E0A6979697C4C" ], "X789E0A6979697C4C" ],
[ "\033[2XIsRCLoop\033[102X", "7.4-7", [ 7, 4, 7 ], 253, 40, "isrcloop", [ "\033[2XIsRCLoop\033[102X", "7.4-7", [ 7, 4, 7 ], 253, 41, "isrcloop",
"X7B03CC577802F4AB" ], "X7B03CC577802F4AB" ],
[ "\033[2XIsLeftNuclearSquareLoop\033[102X", "7.4-8", [ 7, 4, 8 ], 258, 40, [ "\033[2XIsLeftNuclearSquareLoop\033[102X", "7.4-8", [ 7, 4, 8 ], 258, 41,
"isleftnuclearsquareloop", "X819F285887B5EB9E" ], "isleftnuclearsquareloop", "X819F285887B5EB9E" ],
[ "\033[2XIsMiddleNuclearSquareLoop\033[102X", "7.4-9", [ 7, 4, 9 ], 263, [ "\033[2XIsMiddleNuclearSquareLoop\033[102X", "7.4-9", [ 7, 4, 9 ], 263,
40, "ismiddlenuclearsquareloop", "X8474F55681244A8A" ], 41, "ismiddlenuclearsquareloop", "X8474F55681244A8A" ],
[ "\033[2XIsRightNuclearSquareLoop\033[102X", "7.4-10", [ 7, 4, 10 ], 268, [ "\033[2XIsRightNuclearSquareLoop\033[102X", "7.4-10", [ 7, 4, 10 ], 268,
40, "isrightnuclearsquareloop", "X807B3B21825E3076" ], 41, "isrightnuclearsquareloop", "X807B3B21825E3076" ],
[ "\033[2XIsNuclearSquareLoop\033[102X", "7.4-11", [ 7, 4, 11 ], 273, 41, [ "\033[2XIsNuclearSquareLoop\033[102X", "7.4-11", [ 7, 4, 11 ], 273, 42,
"isnuclearsquareloop", "X796650088213229B" ], "isnuclearsquareloop", "X796650088213229B" ],
[ "\033[2XIsFlexible\033[102X", "7.4-12", [ 7, 4, 12 ], 278, 41, [ "\033[2XIsFlexible\033[102X", "7.4-12", [ 7, 4, 12 ], 278, 42,
"isflexible", "X7C32851A7AF1C45F" ], "isflexible", "X7C32851A7AF1C45F" ],
[ "\033[2XIsLeftAlternative\033[102X", "7.4-13", [ 7, 4, 13 ], 283, 41, [ "\033[2XIsLeftAlternative\033[102X", "7.4-13", [ 7, 4, 13 ], 283, 42,
"isleftalternative", "X7DF0196786B9CE08" ], "isleftalternative", "X7DF0196786B9CE08" ],
[ "\033[2XIsRightAlternative\033[102X", "7.4-14", [ 7, 4, 14 ], 288, 41, [ "\033[2XIsRightAlternative\033[102X", "7.4-14", [ 7, 4, 14 ], 288, 42,
"isrightalternative", "X8416FAD87F148F5D" ], "isrightalternative", "X8416FAD87F148F5D" ],
[ "\033[2XIsAlternative\033[102X", "7.4-15", [ 7, 4, 15 ], 293, 41, [ "\033[2XIsAlternative\033[102X", "7.4-15", [ 7, 4, 15 ], 293, 42,
"isalternative", "X8379356E82DB5DDA" ], "isalternative", "X8379356E82DB5DDA" ],
[ "power alternative loop left", "7.5", [ 7, 5, 0 ], 324, 42, [ "power alternative loop left", "7.5", [ 7, 5, 0 ], 324, 43,
"power alternative loop left", "X83A501387E1AC371" ], "power alternative loop left", "X83A501387E1AC371" ],
[ "loop left power alternative", "7.5", [ 7, 5, 0 ], 324, 42, [ "loop left power alternative", "7.5", [ 7, 5, 0 ], 324, 43,
"loop left power alternative", "X83A501387E1AC371" ], "loop left power alternative", "X83A501387E1AC371" ],
[ "power alternative loop right", "7.5", [ 7, 5, 0 ], 324, 42, [ "power alternative loop right", "7.5", [ 7, 5, 0 ], 324, 43,
"power alternative loop right", "X83A501387E1AC371" ], "power alternative loop right", "X83A501387E1AC371" ],
[ "loop right power alternative", "7.5", [ 7, 5, 0 ], 324, 42, [ "loop right power alternative", "7.5", [ 7, 5, 0 ], 324, 43,
"loop right power alternative", "X83A501387E1AC371" ], "loop right power alternative", "X83A501387E1AC371" ],
[ "power alternative loop", "7.5", [ 7, 5, 0 ], 324, 42, [ "power alternative loop", "7.5", [ 7, 5, 0 ], 324, 43,
"power alternative loop", "X83A501387E1AC371" ], "power alternative loop", "X83A501387E1AC371" ],
[ "loop power alternative", "7.5", [ 7, 5, 0 ], 324, 42, [ "loop power alternative", "7.5", [ 7, 5, 0 ], 324, 43,
"loop power alternative", "X83A501387E1AC371" ], "loop power alternative", "X83A501387E1AC371" ],
[ "\033[2XIsLeftPowerAlternative\033[102X", "7.5-1", [ 7, 5, 1 ], 337, 42, [ "\033[2XIsLeftPowerAlternative\033[102X", "7.5-1", [ 7, 5, 1 ], 337, 43,
"isleftpoweralternative", "X875C3DF681B3FAE2" ], "isleftpoweralternative", "X875C3DF681B3FAE2" ],
[ "\033[2XIsRightPowerAlternative\033[102X", "7.5-1", [ 7, 5, 1 ], 337, 42, [ "\033[2XIsRightPowerAlternative\033[102X", "7.5-1", [ 7, 5, 1 ], 337, 43,
"isrightpoweralternative", "X875C3DF681B3FAE2" ], "isrightpoweralternative", "X875C3DF681B3FAE2" ],
[ "\033[2XIsPowerAlternative\033[102X", "7.5-1", [ 7, 5, 1 ], 337, 42, [ "\033[2XIsPowerAlternative\033[102X", "7.5-1", [ 7, 5, 1 ], 337, 43,
"ispoweralternative", "X875C3DF681B3FAE2" ], "ispoweralternative", "X875C3DF681B3FAE2" ],
[ "conjugacy closed loop left", "7.6", [ 7, 6, 0 ], 346, 42, [ "conjugacy closed loop left", "7.6", [ 7, 6, 0 ], 346, 43,
"conjugacy closed loop left", "X8176B2C47A4629CD" ], "conjugacy closed loop left", "X8176B2C47A4629CD" ],
[ "loop left conjugacy closed", "7.6", [ 7, 6, 0 ], 346, 42, [ "loop left conjugacy closed", "7.6", [ 7, 6, 0 ], 346, 43,
"loop left conjugacy closed", "X8176B2C47A4629CD" ], "loop left conjugacy closed", "X8176B2C47A4629CD" ],
[ "conjugacy closed loop right", "7.6", [ 7, 6, 0 ], 346, 42, [ "conjugacy closed loop right", "7.6", [ 7, 6, 0 ], 346, 43,
"conjugacy closed loop right", "X8176B2C47A4629CD" ], "conjugacy closed loop right", "X8176B2C47A4629CD" ],
[ "loop right conjugacy closed", "7.6", [ 7, 6, 0 ], 346, 42, [ "loop right conjugacy closed", "7.6", [ 7, 6, 0 ], 346, 43,
"loop right conjugacy closed", "X8176B2C47A4629CD" ], "loop right conjugacy closed", "X8176B2C47A4629CD" ],
[ "conjugacy closed loop", "7.6", [ 7, 6, 0 ], 346, 42, [ "conjugacy closed loop", "7.6", [ 7, 6, 0 ], 346, 43,
"conjugacy closed loop", "X8176B2C47A4629CD" ], "conjugacy closed loop", "X8176B2C47A4629CD" ],
[ "loop conjugacy closed", "7.6", [ 7, 6, 0 ], 346, 42, [ "loop conjugacy closed", "7.6", [ 7, 6, 0 ], 346, 43,
"loop conjugacy closed", "X8176B2C47A4629CD" ], "loop conjugacy closed", "X8176B2C47A4629CD" ],
[ "\033[2XIsLCCLoop\033[102X", "7.6-1", [ 7, 6, 1 ], 358, 42, "islccloop", [ "\033[2XIsLCCLoop\033[102X", "7.6-1", [ 7, 6, 1 ], 358, 43, "islccloop",
"X784E08CD7B710AF4" ], "X784E08CD7B710AF4" ],
[ "\033[2XIsLeftConjugacyClosedLoop\033[102X", "7.6-1", [ 7, 6, 1 ], 358, [ "\033[2XIsLeftConjugacyClosedLoop\033[102X", "7.6-1", [ 7, 6, 1 ], 358,
42, "isleftconjugacyclosedloop", "X784E08CD7B710AF4" ], 43, "isleftconjugacyclosedloop", "X784E08CD7B710AF4" ],
[ "\033[2XIsRCCLoop\033[102X", "7.6-2", [ 7, 6, 2 ], 364, 42, "isrccloop", [ "\033[2XIsRCCLoop\033[102X", "7.6-2", [ 7, 6, 2 ], 364, 43, "isrccloop",
"X7B3016B47A1A8213" ], "X7B3016B47A1A8213" ],
[ "\033[2XIsRightConjugacyClosedLoop\033[102X", "7.6-2", [ 7, 6, 2 ], 364, [ "\033[2XIsRightConjugacyClosedLoop\033[102X", "7.6-2", [ 7, 6, 2 ], 364,
42, "isrightconjugacyclosedloop", "X7B3016B47A1A8213" ], 43, "isrightconjugacyclosedloop", "X7B3016B47A1A8213" ],
[ "\033[2XIsCCLoop\033[102X", "7.6-3", [ 7, 6, 3 ], 370, 42, "isccloop", [ "\033[2XIsCCLoop\033[102X", "7.6-3", [ 7, 6, 3 ], 370, 43, "isccloop",
"X878B614479DCB83F" ], "X878B614479DCB83F" ],
[ "\033[2XIsConjugacyClosedLoop\033[102X", "7.6-3", [ 7, 6, 3 ], 370, 42, [ "\033[2XIsConjugacyClosedLoop\033[102X", "7.6-3", [ 7, 6, 3 ], 370, 43,
"isconjugacyclosedloop", "X878B614479DCB83F" ], "isconjugacyclosedloop", "X878B614479DCB83F" ],
[ "\033[2XIsOsbornLoop\033[102X", "7.6-4", [ 7, 6, 4 ], 376, 42, [ "\033[2XIsOsbornLoop\033[102X", "7.6-4", [ 7, 6, 4 ], 376, 43,
"isosbornloop", "X8655956878205FC1" ], "isosbornloop", "X8655956878205FC1" ],
[ "Osborn loop", "7.6-4", [ 7, 6, 4 ], 376, 42, "osborn loop", [ "Osborn loop", "7.6-4", [ 7, 6, 4 ], 376, 43, "osborn loop",
"X8655956878205FC1" ], "X8655956878205FC1" ],
[ "loop Osborn", "7.6-4", [ 7, 6, 4 ], 376, 42, "loop osborn", [ "loop Osborn", "7.6-4", [ 7, 6, 4 ], 376, 43, "loop osborn",
"X8655956878205FC1" ], "X8655956878205FC1" ],
[ "automorphic loop left", "7.7", [ 7, 7, 0 ], 384, 43, [ "automorphic loop left", "7.7", [ 7, 7, 0 ], 384, 44,
"automorphic loop left", "X793B22EA8643C667" ], "automorphic loop left", "X793B22EA8643C667" ],
[ "loop left automorphic", "7.7", [ 7, 7, 0 ], 384, 43, [ "loop left automorphic", "7.7", [ 7, 7, 0 ], 384, 44,
"loop left automorphic", "X793B22EA8643C667" ], "loop left automorphic", "X793B22EA8643C667" ],
[ "automorphic loop middle", "7.7", [ 7, 7, 0 ], 384, 43, [ "automorphic loop middle", "7.7", [ 7, 7, 0 ], 384, 44,
"automorphic loop middle", "X793B22EA8643C667" ], "automorphic loop middle", "X793B22EA8643C667" ],
[ "loop middle automorphic", "7.7", [ 7, 7, 0 ], 384, 43, [ "loop middle automorphic", "7.7", [ 7, 7, 0 ], 384, 44,
"loop middle automorphic", "X793B22EA8643C667" ], "loop middle automorphic", "X793B22EA8643C667" ],
[ "automorphic loop right", "7.7", [ 7, 7, 0 ], 384, 43, [ "automorphic loop right", "7.7", [ 7, 7, 0 ], 384, 44,
"automorphic loop right", "X793B22EA8643C667" ], "automorphic loop right", "X793B22EA8643C667" ],
[ "loop right automorphic", "7.7", [ 7, 7, 0 ], 384, 43, [ "loop right automorphic", "7.7", [ 7, 7, 0 ], 384, 44,
"loop right automorphic", "X793B22EA8643C667" ], "loop right automorphic", "X793B22EA8643C667" ],
[ "automorphic loop", "7.7", [ 7, 7, 0 ], 384, 43, "automorphic loop", [ "automorphic loop", "7.7", [ 7, 7, 0 ], 384, 44, "automorphic loop",
"X793B22EA8643C667" ], "X793B22EA8643C667" ],
[ "loop automorphic", "7.7", [ 7, 7, 0 ], 384, 43, "loop automorphic", [ "loop automorphic", "7.7", [ 7, 7, 0 ], 384, 44, "loop automorphic",
"X793B22EA8643C667" ], "X793B22EA8643C667" ],
[ "\033[2XIsLeftAutomorphicLoop\033[102X", "7.7-1", [ 7, 7, 1 ], 425, 43, [ "\033[2XIsLeftAutomorphicLoop\033[102X", "7.7-1", [ 7, 7, 1 ], 425, 44,
"isleftautomorphicloop", "X7F063914804659F1" ], "isleftautomorphicloop", "X7F063914804659F1" ],
[ "\033[2XIsLeftALoop\033[102X", "7.7-1", [ 7, 7, 1 ], 425, 43, [ "\033[2XIsLeftALoop\033[102X", "7.7-1", [ 7, 7, 1 ], 425, 44,
"isleftaloop", "X7F063914804659F1" ], "isleftaloop", "X7F063914804659F1" ],
[ "\033[2XIsMiddleAutomorphicLoop\033[102X", "7.7-2", [ 7, 7, 2 ], 431, 43, [ "\033[2XIsMiddleAutomorphicLoop\033[102X", "7.7-2", [ 7, 7, 2 ], 431, 44,
"ismiddleautomorphicloop", "X7DFE830584A769E5" ], "ismiddleautomorphicloop", "X7DFE830584A769E5" ],
[ "\033[2XIsMiddleALoop\033[102X", "7.7-2", [ 7, 7, 2 ], 431, 43, [ "\033[2XIsMiddleALoop\033[102X", "7.7-2", [ 7, 7, 2 ], 431, 44,
"ismiddlealoop", "X7DFE830584A769E5" ], "ismiddlealoop", "X7DFE830584A769E5" ],
[ "\033[2XIsRightAutomorphicLoop\033[102X", "7.7-3", [ 7, 7, 3 ], 437, 44, [ "\033[2XIsRightAutomorphicLoop\033[102X", "7.7-3", [ 7, 7, 3 ], 437, 45,
"isrightautomorphicloop", "X7EA9165A87F99E35" ], "isrightautomorphicloop", "X7EA9165A87F99E35" ],
[ "\033[2XIsRightALoop\033[102X", "7.7-3", [ 7, 7, 3 ], 437, 44, [ "\033[2XIsRightALoop\033[102X", "7.7-3", [ 7, 7, 3 ], 437, 45,
"isrightaloop", "X7EA9165A87F99E35" ], "isrightaloop", "X7EA9165A87F99E35" ],
[ "\033[2XIsAutomorphicLoop\033[102X", "7.7-4", [ 7, 7, 4 ], 443, 44, [ "\033[2XIsAutomorphicLoop\033[102X", "7.7-4", [ 7, 7, 4 ], 443, 45,
"isautomorphicloop", "X7899603184CF13FD" ], "isautomorphicloop", "X7899603184CF13FD" ],
[ "\033[2XIsALoop\033[102X", "7.7-4", [ 7, 7, 4 ], 443, 44, "isaloop", [ "\033[2XIsALoop\033[102X", "7.7-4", [ 7, 7, 4 ], 443, 45, "isaloop",
"X7899603184CF13FD" ], "X7899603184CF13FD" ],
[ "\033[2XIsCodeLoop\033[102X", "7.8-1", [ 7, 8, 1 ], 454, 44, [ "\033[2XIsCodeLoop\033[102X", "7.8-1", [ 7, 8, 1 ], 454, 45,
"iscodeloop", "X790FA1188087D5C1" ], "iscodeloop", "X790FA1188087D5C1" ],
[ "code loop", "7.8-1", [ 7, 8, 1 ], 454, 44, "code loop", [ "code loop", "7.8-1", [ 7, 8, 1 ], 454, 45, "code loop",
"X790FA1188087D5C1" ], "X790FA1188087D5C1" ],
[ "loop code", "7.8-1", [ 7, 8, 1 ], 454, 44, "loop code", [ "loop code", "7.8-1", [ 7, 8, 1 ], 454, 45, "loop code",
"X790FA1188087D5C1" ], "X790FA1188087D5C1" ],
[ "\033[2XIsSteinerLoop\033[102X", "7.8-2", [ 7, 8, 2 ], 462, 44, [ "\033[2XIsSteinerLoop\033[102X", "7.8-2", [ 7, 8, 2 ], 462, 45,
"issteinerloop", "X793600C9801F4F62" ], "issteinerloop", "X793600C9801F4F62" ],
[ "Steiner loop", "7.8-2", [ 7, 8, 2 ], 462, 44, "steiner loop", [ "Steiner loop", "7.8-2", [ 7, 8, 2 ], 462, 45, "steiner loop",
"X793600C9801F4F62" ], "X793600C9801F4F62" ],
[ "loop Steiner", "7.8-2", [ 7, 8, 2 ], 462, 44, "loop steiner", [ "loop Steiner", "7.8-2", [ 7, 8, 2 ], 462, 45, "loop steiner",
"X793600C9801F4F62" ], "X793600C9801F4F62" ],
[ "\033[2XIsLeftBruckLoop\033[102X", "7.8-3", [ 7, 8, 3 ], 470, 44, [ "\033[2XIsLeftBruckLoop\033[102X", "7.8-3", [ 7, 8, 3 ], 470, 45,
"isleftbruckloop", "X85F1BD4280E44F5B" ], "isleftbruckloop", "X85F1BD4280E44F5B" ],
[ "\033[2XIsLeftKLoop\033[102X", "7.8-3", [ 7, 8, 3 ], 470, 44, [ "\033[2XIsLeftKLoop\033[102X", "7.8-3", [ 7, 8, 3 ], 470, 45,
"isleftkloop", "X85F1BD4280E44F5B" ], "isleftkloop", "X85F1BD4280E44F5B" ],
[ "Bruck loop left", "7.8-3", [ 7, 8, 3 ], 470, 44, "bruck loop left", [ "Bruck loop left", "7.8-3", [ 7, 8, 3 ], 470, 45, "bruck loop left",
"X85F1BD4280E44F5B" ], "X85F1BD4280E44F5B" ],
[ "loop left Bruck", "7.8-3", [ 7, 8, 3 ], 470, 44, "loop left bruck", [ "loop left Bruck", "7.8-3", [ 7, 8, 3 ], 470, 45, "loop left bruck",
"X85F1BD4280E44F5B" ], "X85F1BD4280E44F5B" ],
[ "K loop left", "7.8-3", [ 7, 8, 3 ], 470, 44, "k loop left", [ "K loop left", "7.8-3", [ 7, 8, 3 ], 470, 45, "k loop left",
"X85F1BD4280E44F5B" ], "X85F1BD4280E44F5B" ],
[ "loop left K", "7.8-3", [ 7, 8, 3 ], 470, 44, "loop left k", [ "loop left K", "7.8-3", [ 7, 8, 3 ], 470, 45, "loop left k",
"X85F1BD4280E44F5B" ], "X85F1BD4280E44F5B" ],
[ "\033[2XIsRightBruckLoop\033[102X", "7.8-4", [ 7, 8, 4 ], 480, 44, [ "\033[2XIsRightBruckLoop\033[102X", "7.8-4", [ 7, 8, 4 ], 480, 45,
"isrightbruckloop", "X857B373E7B4E0519" ], "isrightbruckloop", "X857B373E7B4E0519" ],
[ "\033[2XIsRightKLoop\033[102X", "7.8-4", [ 7, 8, 4 ], 480, 44, [ "\033[2XIsRightKLoop\033[102X", "7.8-4", [ 7, 8, 4 ], 480, 45,
"isrightkloop", "X857B373E7B4E0519" ], "isrightkloop", "X857B373E7B4E0519" ],
[ "Bruck loop right", "7.8-4", [ 7, 8, 4 ], 480, 44, "bruck loop right", [ "Bruck loop right", "7.8-4", [ 7, 8, 4 ], 480, 45, "bruck loop right",
"X857B373E7B4E0519" ], "X857B373E7B4E0519" ],
[ "loop right Bruck", "7.8-4", [ 7, 8, 4 ], 480, 44, "loop right bruck", [ "loop right Bruck", "7.8-4", [ 7, 8, 4 ], 480, 45, "loop right bruck",
"X857B373E7B4E0519" ], "X857B373E7B4E0519" ],
[ "K loop right", "7.8-4", [ 7, 8, 4 ], 480, 44, "k loop right", [ "K loop right", "7.8-4", [ 7, 8, 4 ], 480, 45, "k loop right",
"X857B373E7B4E0519" ], "X857B373E7B4E0519" ],
[ "loop right K", "7.8-4", [ 7, 8, 4 ], 480, 44, "loop right k", [ "loop right K", "7.8-4", [ 7, 8, 4 ], 480, 45, "loop right k",
"X857B373E7B4E0519" ], "X857B373E7B4E0519" ],
[ "\033[2XAssociatedLeftBruckLoop\033[102X", "8.1-1", [ 8, 1, 1 ], 10, 45, [ "\033[2XAssociatedLeftBruckLoop\033[102X", "8.1-1", [ 8, 1, 1 ], 10, 46,
"associatedleftbruckloop", "X8664CA927DD73DBE" ], "associatedleftbruckloop", "X8664CA927DD73DBE" ],
[ "\033[2XAssociatedRightBruckLoop\033[102X", "8.1-1", [ 8, 1, 1 ], 10, 45, [ "\033[2XAssociatedRightBruckLoop\033[102X", "8.1-1", [ 8, 1, 1 ], 10, 46,
"associatedrightbruckloop", "X8664CA927DD73DBE" ], "associatedrightbruckloop", "X8664CA927DD73DBE" ],
[ "loop left Bol", "8.1-1", [ 8, 1, 1 ], 10, 45, "loop left bol", [ "loop left Bol", "8.1-1", [ 8, 1, 1 ], 10, 46, "loop left bol",
"X8664CA927DD73DBE" ], "X8664CA927DD73DBE" ],
[ "Bol loop left", "8.1-1", [ 8, 1, 1 ], 10, 45, "bol loop left", [ "Bol loop left", "8.1-1", [ 8, 1, 1 ], 10, 46, "bol loop left",
"X8664CA927DD73DBE" ], "X8664CA927DD73DBE" ],
[ "Bruck loop associated left", "8.1-1", [ 8, 1, 1 ], 10, 45, [ "Bruck loop associated left", "8.1-1", [ 8, 1, 1 ], 10, 46,
"bruck loop associated left", "X8664CA927DD73DBE" ], "bruck loop associated left", "X8664CA927DD73DBE" ],
[ "loop associated left Bruck", "8.1-1", [ 8, 1, 1 ], 10, 45, [ "loop associated left Bruck", "8.1-1", [ 8, 1, 1 ], 10, 46,
"loop associated left bruck", "X8664CA927DD73DBE" ], "loop associated left bruck", "X8664CA927DD73DBE" ],
[ "\033[2XIsExactGroupFactorization\033[102X", "8.1-2", [ 8, 1, 2 ], 26, [ "\033[2XIsExactGroupFactorization\033[102X", "8.1-2", [ 8, 1, 2 ], 26,
45, "isexactgroupfactorization", "X82FC16F386CE11F1" ], 46, "isexactgroupfactorization", "X82FC16F386CE11F1" ],
[ "exact group factorization", "8.1-2", [ 8, 1, 2 ], 26, 45, [ "exact group factorization", "8.1-2", [ 8, 1, 2 ], 26, 46,
"exact group factorization", "X82FC16F386CE11F1" ], "exact group factorization", "X82FC16F386CE11F1" ],
[ "\033[2XRightBolLoopByExactGroupFactorization\033[102X", "8.1-3", [ "\033[2XRightBolLoopByExactGroupFactorization\033[102X", "8.1-3",
[ 8, 1, 3 ], 35, 45, "rightbolloopbyexactgroupfactorization", [ 8, 1, 3 ], 35, 46, "rightbolloopbyexactgroupfactorization",
"X7DCA64807F899127" ], "X7DCA64807F899127" ],
[ "modification Moufang", "8.2", [ 8, 2, 0 ], 47, 46, [ "modification Moufang", "8.2", [ 8, 2, 0 ], 47, 47,
"modification moufang", "X819F82737C2A860D" ], "modification moufang", "X819F82737C2A860D" ],
[ "\033[2XLoopByCyclicModification\033[102X", "8.2-1", [ 8, 2, 1 ], 57, 46, [ "\033[2XLoopByCyclicModification\033[102X", "8.2-1", [ 8, 2, 1 ], 57, 47,
"loopbycyclicmodification", "X7B3165C083709831" ], "loopbycyclicmodification", "X7B3165C083709831" ],
[ "modification cyclic", "8.2-1", [ 8, 2, 1 ], 57, 46, [ "modification cyclic", "8.2-1", [ 8, 2, 1 ], 57, 47,
"modification cyclic", "X7B3165C083709831" ], "modification cyclic", "X7B3165C083709831" ],
[ "\033[2XLoopByDihedralModification\033[102X", "8.2-2", [ 8, 2, 2 ], 70, [ "\033[2XLoopByDihedralModification\033[102X", "8.2-2", [ 8, 2, 2 ], 70,
46, "loopbydihedralmodification", "X7D7717C587BC2D1E" ], 47, "loopbydihedralmodification", "X7D7717C587BC2D1E" ],
[ "modification dihedral", "8.2-2", [ 8, 2, 2 ], 70, 46, [ "modification dihedral", "8.2-2", [ 8, 2, 2 ], 70, 47,
"modification dihedral", "X7D7717C587BC2D1E" ], "modification dihedral", "X7D7717C587BC2D1E" ],
[ "\033[2XLoopMG2\033[102X", "8.2-3", [ 8, 2, 3 ], 86, 46, "loopmg2", [ "\033[2XLoopMG2\033[102X", "8.2-3", [ 8, 2, 3 ], 86, 47, "loopmg2",
"X7CC6CDB786E9BBA0" ], "X7CC6CDB786E9BBA0" ],
[ "Chein loop", "8.2-3", [ 8, 2, 3 ], 86, 46, "chein loop", [ "Chein loop", "8.2-3", [ 8, 2, 3 ], 86, 47, "chein loop",
"X7CC6CDB786E9BBA0" ], "X7CC6CDB786E9BBA0" ],
[ "loop Chein", "8.2-3", [ 8, 2, 3 ], 86, 46, "loop chein", [ "loop Chein", "8.2-3", [ 8, 2, 3 ], 86, 47, "loop chein",
"X7CC6CDB786E9BBA0" ], "X7CC6CDB786E9BBA0" ],
[ "group with triality", "8.3", [ 8, 3, 0 ], 98, 46, "group with triality", [ "group with triality", "8.3", [ 8, 3, 0 ], 98, 47, "group with triality",
"X83E73A767D79FAFD" ], "X83E73A767D79FAFD" ],
[ "\033[2XTrialityPermGroup\033[102X", "8.3-1", [ 8, 3, 1 ], 113, 47, [ "\033[2XTrialityPermGroup\033[102X", "8.3-1", [ 8, 3, 1 ], 113, 48,
"trialitypermgroup", "X7DB4DE647F6F56F0" ], "trialitypermgroup", "X7DB4DE647F6F56F0" ],
[ "\033[2XTrialityPcGroup\033[102X", "8.3-2", [ 8, 3, 2 ], 120, 47, [ "\033[2XTrialityPcGroup\033[102X", "8.3-2", [ 8, 3, 2 ], 120, 48,
"trialitypcgroup", "X82CC977085DFDFE8" ], "trialitypcgroup", "X82CC977085DFDFE8" ],
[ "\033[2XAllLoopTablesInGroup\033[102X", "8.4-1", [ 8, 4, 1 ], 146, 47, [ "\033[2XAllLoopTablesInGroup\033[102X", "8.4-1", [ 8, 4, 1 ], 146, 48,
"alllooptablesingroup", "X804F40087DD1225D" ], "alllooptablesingroup", "X804F40087DD1225D" ],
[ "\033[2XAllProperLoopTablesInGroup\033[102X", "8.4-2", [ 8, 4, 2 ], 152, [ "\033[2XAllProperLoopTablesInGroup\033[102X", "8.4-2", [ 8, 4, 2 ], 152,
47, "allproperlooptablesingroup", "X7854C8E382DC8E8B" ], 48, "allproperlooptablesingroup", "X7854C8E382DC8E8B" ],
[ "\033[2XOneLoopTableInGroup\033[102X", "8.4-3", [ 8, 4, 3 ], 158, 47, [ "\033[2XOneLoopTableInGroup\033[102X", "8.4-3", [ 8, 4, 3 ], 158, 48,
"onelooptableingroup", "X7BFFC66A824BA6AA" ], "onelooptableingroup", "X7BFFC66A824BA6AA" ],
[ "\033[2XOneProperLoopTableInGroup\033[102X", "8.4-4", [ 8, 4, 4 ], 164, [ "\033[2XOneProperLoopTableInGroup\033[102X", "8.4-4", [ 8, 4, 4 ], 164,
48, "oneproperlooptableingroup", "X84C5A76585B335FF" ], 49, "oneproperlooptableingroup", "X84C5A76585B335FF" ],
[ "\033[2XAllLoopsWithMltGroup\033[102X", "8.4-5", [ 8, 4, 5 ], 170, 48, [ "\033[2XAllLoopsWithMltGroup\033[102X", "8.4-5", [ 8, 4, 5 ], 170, 49,
"allloopswithmltgroup", "X7E5F1C2879358EEF" ], "allloopswithmltgroup", "X7E5F1C2879358EEF" ],
[ "\033[2XOneLoopWithMltGroup\033[102X", "8.4-6", [ 8, 4, 6 ], 176, 48, [ "\033[2XOneLoopWithMltGroup\033[102X", "8.4-6", [ 8, 4, 6 ], 176, 49,
"oneloopwithmltgroup", "X8266DE05824226E6" ], "oneloopwithmltgroup", "X8266DE05824226E6" ],
[ "\033[2XLibraryLoop\033[102X", "9.1-1", [ 9, 1, 1 ], 31, 49, [ "\033[2XLibraryLoop\033[102X", "9.1-1", [ 9, 1, 1 ], 31, 50,
"libraryloop", "X849865D6786EEF9B" ], "libraryloop", "X849865D6786EEF9B" ],
[ "\033[2XMyLibraryLoop\033[102X", "9.1-2", [ 9, 1, 2 ], 36, 49, [ "\033[2XMyLibraryLoop\033[102X", "9.1-2", [ 9, 1, 2 ], 36, 50,
"mylibraryloop", "X78C4B8757902D49F" ], "mylibraryloop", "X78C4B8757902D49F" ],
[ "\033[2XDisplayLibraryInfo\033[102X", "9.1-3", [ 9, 1, 3 ], 46, 50, [ "\033[2XDisplayLibraryInfo\033[102X", "9.1-3", [ 9, 1, 3 ], 46, 51,
"displaylibraryinfo", "X7A64372E81E713B4" ], "displaylibraryinfo", "X7A64372E81E713B4" ],
[ "\033[2XLeftBolLoop\033[102X", "9.2-1", [ 9, 2, 1 ], 67, 50, [ "\033[2XLeftBolLoop\033[102X", "9.2-1", [ 9, 2, 1 ], 67, 51,
"leftbolloop", "X7EE99F647C537994" ], "leftbolloop", "X7EE99F647C537994" ],
[ "\033[2XRightBolLoop\033[102X", "9.2-2", [ 9, 2, 2 ], 72, 50, [ "\033[2XRightBolLoop\033[102X", "9.2-2", [ 9, 2, 2 ], 72, 51,
"rightbolloop", "X8774304282654C58" ], "rightbolloop", "X8774304282654C58" ],
[ "\033[2XMoufangLoop\033[102X", "9.3-1", [ 9, 3, 1 ], 86, 50, [ "\033[2XLeftBruckLoop\033[102X", "9.3-1", [ 9, 3, 1 ], 92, 51,
"leftbruckloop", "X8290B01780F0FCD3" ],
[ "\033[2XRightBruckLoop\033[102X", "9.3-2", [ 9, 3, 2 ], 97, 51,
"rightbruckloop", "X798DD7CF871F648F" ],
[ "\033[2XMoufangLoop\033[102X", "9.4-1", [ 9, 4, 1 ], 108, 52,
"moufangloop", "X81E82098822543EE" ], "moufangloop", "X81E82098822543EE" ],
[ "octonion loop", "9.3-1", [ 9, 3, 1 ], 86, 50, "octonion loop", [ "octonion loop", "9.4-1", [ 9, 4, 1 ], 108, 52, "octonion loop",
"X81E82098822543EE" ], "X81E82098822543EE" ],
[ "loop octonion", "9.3-1", [ 9, 3, 1 ], 86, 50, "loop octonion", [ "loop octonion", "9.4-1", [ 9, 4, 1 ], 108, 52, "loop octonion",
"X81E82098822543EE" ], "X81E82098822543EE" ],
[ "\033[2XCodeLoop\033[102X", "9.4-1", [ 9, 4, 1 ], 117, 51, "codeloop", [ "\033[2XCodeLoop\033[102X", "9.5-1", [ 9, 5, 1 ], 139, 52, "codeloop",
"X7DB4D3B27BB4D7EE" ], "X7DB4D3B27BB4D7EE" ],
[ "\033[2XSteinerLoop\033[102X", "9.5-1", [ 9, 5, 1 ], 144, 51, [ "\033[2XSteinerLoop\033[102X", "9.6-1", [ 9, 6, 1 ], 166, 53,
"steinerloop", "X87C235457E859AF4" ], "steinerloop", "X87C235457E859AF4" ],
[ "\033[2XRCCLoop\033[102X", "9.6-1", [ 9, 6, 1 ], 173, 52, "rccloop", [ "\033[2XRCCLoop\033[102X", "9.7-1", [ 9, 7, 1 ], 195, 53, "rccloop",
"X806B2DE67990E42F" ], "X806B2DE67990E42F" ],
[ "\033[2XRightConjugacyClosedLoop\033[102X", "9.6-1", [ 9, 6, 1 ], 173, [ "\033[2XRightConjugacyClosedLoop\033[102X", "9.7-1", [ 9, 7, 1 ], 195,
52, "rightconjugacyclosedloop", "X806B2DE67990E42F" ], 53, "rightconjugacyclosedloop", "X806B2DE67990E42F" ],
[ "\033[2XLCCLoop\033[102X", "9.6-2", [ 9, 6, 2 ], 180, 52, "lccloop", [ "\033[2XLCCLoop\033[102X", "9.7-2", [ 9, 7, 2 ], 202, 53, "lccloop",
"X80AB8B107D55FB19" ], "X80AB8B107D55FB19" ],
[ "\033[2XLeftConjugacyClosedLoop\033[102X", "9.6-2", [ 9, 6, 2 ], 180, 52, [ "\033[2XLeftConjugacyClosedLoop\033[102X", "9.7-2", [ 9, 7, 2 ], 202, 53,
"leftconjugacyclosedloop", "X80AB8B107D55FB19" ], "leftconjugacyclosedloop", "X80AB8B107D55FB19" ],
[ "\033[2XCCLoop\033[102X", "9.6-3", [ 9, 6, 3 ], 217, 52, "ccloop", [ "\033[2XCCLoop\033[102X", "9.7-3", [ 9, 7, 3 ], 241, 54, "ccloop",
"X798BC601843E8916" ], "X798BC601843E8916" ],
[ "\033[2XConjugacyClosedLoop\033[102X", "9.6-3", [ 9, 6, 3 ], 217, 52, [ "\033[2XConjugacyClosedLoop\033[102X", "9.7-3", [ 9, 7, 3 ], 241, 54,
"conjugacyclosedloop", "X798BC601843E8916" ], "conjugacyclosedloop", "X798BC601843E8916" ],
[ "\033[2XSmallLoop\033[102X", "9.7-1", [ 9, 7, 1 ], 230, 53, "smallloop", [ "\033[2XSmallLoop\033[102X", "9.8-1", [ 9, 8, 1 ], 254, 54, "smallloop",
"X7C6EE23E84CD87D3" ], "X7C6EE23E84CD87D3" ],
[ "Paige loop", "9.8", [ 9, 8, 0 ], 235, 53, "paige loop", [ "Paige loop", "9.9", [ 9, 9, 0 ], 259, 54, "paige loop",
"X8135C8FD8714C606" ], "X8135C8FD8714C606" ],
[ "loop Paige", "9.8", [ 9, 8, 0 ], 235, 53, "loop paige", [ "loop Paige", "9.9", [ 9, 9, 0 ], 259, 54, "loop paige",
"X8135C8FD8714C606" ], "X8135C8FD8714C606" ],
[ "\033[2XPaigeLoop\033[102X", "9.8-1", [ 9, 8, 1 ], 244, 53, "paigeloop", [ "\033[2XPaigeLoop\033[102X", "9.9-1", [ 9, 9, 1 ], 268, 54, "paigeloop",
"X7FCF4D6B7AD66D74" ], "X7FCF4D6B7AD66D74" ],
[ "\033[2XNilpotentLoop\033[102X", "9.9-1", [ 9, 9, 1 ], 261, 53, [ "\033[2XNilpotentLoop\033[102X", "9.10-1", [ 9, 10, 1 ], 285, 54,
"nilpotentloop", "X7A9C960D86E2AD28" ], "nilpotentloop", "X7A9C960D86E2AD28" ],
[ "\033[2XAutomorphicLoop\033[102X", "9.10-1", [ 9, 10, 1 ], 278, 53, [ "\033[2XAutomorphicLoop\033[102X", "9.11-1", [ 9, 11, 1 ], 304, 55,
"automorphicloop", "X784FFA9E7FDA9F43" ], "automorphicloop", "X784FFA9E7FDA9F43" ],
[ "sedenion loop", "9.11", [ 9, 11, 0 ], 283, 54, "sedenion loop", [ "sedenion loop", "9.12", [ 9, 12, 0 ], 309, 55, "sedenion loop",
"X843BD73F788049F7" ], "X843BD73F788049F7" ],
[ "loop sedenion", "9.11", [ 9, 11, 0 ], 283, 54, "loop sedenion", [ "loop sedenion", "9.12", [ 9, 12, 0 ], 309, 55, "loop sedenion",
"X843BD73F788049F7" ], "X843BD73F788049F7" ],
[ "\033[2XInterestingLoop\033[102X", "9.11-1", [ 9, 11, 1 ], 293, 54, [ "\033[2XInterestingLoop\033[102X", "9.12-1", [ 9, 12, 1 ], 319, 55,
"interestingloop", "X87F24AD3811910D3" ], "interestingloop", "X87F24AD3811910D3" ],
[ "\033[2XItpSmallLoop\033[102X", "9.12-1", [ 9, 12, 1 ], 306, 54, [ "\033[2XItpSmallLoop\033[102X", "9.13-1", [ 9, 13, 1 ], 332, 55,
"itpsmallloop", "X850C4C01817A098D" ] ] "itpsmallloop", "X850C4C01817A098D" ] ]
); );

6
etc/gapdoc.txt
View File

@ -22,7 +22,7 @@ chooser.html
When files are ready, run the following in GAP: When files are ready, run the following in GAP:
# path to files, change as needed # path to files, change as needed
path := Directory("c:/cygwin64/opt/gap4r7/pkg/loops/doc");; path := Directory("c:/cygwin64/opt/gap4r8/pkg/loops/doc");;
main := "loops.xml";; main := "loops.xml";;
files := [];; files := [];;
bookname := "loops";; bookname := "loops";;
@ -52,7 +52,7 @@ GAPDoc2HTMLPrintHTMLFiles(h, path);
# h := GAPDoc2HTML(r, path );; # h := GAPDoc2HTML(r, path );;
# GAPDoc2HTMLPrintHTMLFiles(h, path); # GAPDoc2HTMLPrintHTMLFiles(h, path);
# now produce .ps, .dvi from .tex, # now produce .ps from .tex
# and copy loops.* as manual.* for extensions pdf, ps, dvi # and copy loops.* as manual.* for extensions pdf, ps
# delete auxiliary files # delete auxiliary files

4
gap/banner.g
View File

@ -2,7 +2,7 @@
## ##
#A banner.g loops G. P. Nagy / P. Vojtechovsky #A banner.g loops G. P. Nagy / P. Vojtechovsky
## ##
#H @(#)$Id: banner.g, v 3.3.0 2016/09/21 gap Exp $ #H @(#)$Id: banner.g, v 3.4.0 2017/10/27 gap Exp $
## ##
#Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary), #Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary),
#Y P. Vojtechovsky (University of Denver, USA) #Y P. Vojtechovsky (University of Denver, USA)
@ -12,7 +12,7 @@ if not QUIET and BANNER then
Print( Print(
" ______________________________________________________\n", " ______________________________________________________\n",
" LOOPS: Computing with quasigroups and loops in GAP \n", " LOOPS: Computing with quasigroups and loops in GAP \n",
" version 3.3.0 \n", " version 3.4.0 \n",
" Gabor P. Nagy & Petr Vojtechovsky \n", " Gabor P. Nagy & Petr Vojtechovsky \n",
" nagyg@math.u-szeged.hu petr@math.du.edu \n", " nagyg@math.u-szeged.hu petr@math.du.edu \n",
" ------------------------------------------------------\n", " ------------------------------------------------------\n",

12
gap/classes.gi
View File

@ -2,7 +2,7 @@
## ##
#W classes.gi Testing properties/varieties [loops] #W classes.gi Testing properties/varieties [loops]
## ##
#H @(#)$Id: classes.gi, v 3.3.0 2016/10/26 gap Exp $ #H @(#)$Id: classes.gi, v 3.4.0 2017/10/26 gap Exp $
## ##
#Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary), #Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary),
#Y P. Vojtechovsky (University of Denver, USA) #Y P. Vojtechovsky (University of Denver, USA)
@ -887,16 +887,14 @@ end);
InstallMethod( IsALoop, "for loop", InstallMethod( IsALoop, "for loop",
[ IsLoop ], [ IsLoop ],
function( Q ) function( Q )
return IsLeftALoop(Q) and IsRightALoop(Q) and IsMiddleALoop(Q); return IsRightALoop(Q) and IsMiddleALoop(Q);
# Theorem: rigth A-loop + middle A-loop implies left A-loop
end); end);
# implies # implies
InstallTrueMethod( IsLeftALoop, IsALoop ); InstallTrueMethod( IsLeftALoop, IsALoop );
InstallTrueMethod( IsRightALoop, IsALoop ); InstallTrueMethod( IsRightALoop, IsALoop );
InstallTrueMethod( IsMiddleALoop, IsALoop ); InstallTrueMethod( IsMiddleALoop, IsALoop );
InstallTrueMethod( IsMiddleALoop, IsCommutative );
InstallTrueMethod( IsALoop, IsLeftALoop and IsCommutative );
InstallTrueMethod( IsALoop, IsRightALoop and IsCommutative );
InstallTrueMethod( IsLeftALoop, IsRightALoop and HasAntiautomorphicInverseProperty ); InstallTrueMethod( IsLeftALoop, IsRightALoop and HasAntiautomorphicInverseProperty );
InstallTrueMethod( IsRightALoop, IsLeftALoop and HasAntiautomorphicInverseProperty ); InstallTrueMethod( IsRightALoop, IsLeftALoop and HasAntiautomorphicInverseProperty );
InstallTrueMethod( IsFlexible, IsMiddleALoop ); InstallTrueMethod( IsFlexible, IsMiddleALoop );
@ -909,8 +907,12 @@ InstallTrueMethod( IsMoufangLoop, IsALoop and HasRightInverseProperty );
InstallTrueMethod( IsMoufangLoop, IsALoop and HasWeakInverseProperty ); InstallTrueMethod( IsMoufangLoop, IsALoop and HasWeakInverseProperty );
# is implied by # is implied by
InstallTrueMethod( IsMiddleALoop, IsCommutative );
InstallTrueMethod( IsLeftALoop, IsLeftBruckLoop ); InstallTrueMethod( IsLeftALoop, IsLeftBruckLoop );
InstallTrueMethod( IsLeftALoop, IsLCCLoop ); InstallTrueMethod( IsLeftALoop, IsLCCLoop );
InstallTrueMethod( IsRightALoop, IsRightBruckLoop ); InstallTrueMethod( IsRightALoop, IsRightBruckLoop );
InstallTrueMethod( IsRightALoop, IsRCCLoop ); InstallTrueMethod( IsRightALoop, IsRCCLoop );
InstallTrueMethod( IsALoop, IsCommutative and IsMoufangLoop ); InstallTrueMethod( IsALoop, IsCommutative and IsMoufangLoop );
InstallTrueMethod( IsALoop, IsLeftALoop and IsMiddleALoop );
InstallTrueMethod( IsALoop, IsRightALoop and IsMiddleALoop );
InstallTrueMethod( IsALoop, IsAssociative );

5
gap/examples.gd
View File

@ -2,7 +2,7 @@
## ##
#W examples.gd Examples [loops] #W examples.gd Examples [loops]
## ##
#H @(#)$Id: examples.gd, v 3.1.0 2015/09/23 gap Exp $ #H @(#)$Id: examples.gd, v 3.4.0 2015/09/23 gap Exp $
## ##
#Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary), #Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary),
#Y P. Vojtechovsky (University of Denver, USA) #Y P. Vojtechovsky (University of Denver, USA)
@ -36,6 +36,8 @@ DeclareGlobalFunction( "SmallLoop" );
DeclareGlobalFunction( "InterestingLoop" ); DeclareGlobalFunction( "InterestingLoop" );
DeclareGlobalFunction( "NilpotentLoop" ); DeclareGlobalFunction( "NilpotentLoop" );
DeclareGlobalFunction( "AutomorphicLoop" ); DeclareGlobalFunction( "AutomorphicLoop" );
DeclareGlobalFunction( "LeftBruckLoop" );
DeclareGlobalFunction( "RightBruckLoop" );
# up to isotopism # up to isotopism
@ -52,3 +54,4 @@ DeclareGlobalFunction( "LOOPS_ActivateRCCLoop" );
DeclareGlobalFunction( "LOOPS_ActivateCCLoop" ); DeclareGlobalFunction( "LOOPS_ActivateCCLoop" );
DeclareGlobalFunction( "LOOPS_ActivateNilpotentLoop" ); DeclareGlobalFunction( "LOOPS_ActivateNilpotentLoop" );
DeclareGlobalFunction( "LOOPS_ActivateAutomorphicLoop" ); DeclareGlobalFunction( "LOOPS_ActivateAutomorphicLoop" );
DeclareGlobalFunction( "LOOPS_ActivateRightBruckLoop" );

211
gap/examples.gi
View File

@ -2,7 +2,7 @@
## ##
#W examples.gi Examples [loops] #W examples.gi Examples [loops]
## ##
#H @(#)$Id: examples.gi, v 3.3.0 2016/10/19 gap Exp $ #H @(#)$Id: examples.gi, v 3.4.0 2017/10/23 gap Exp $
## ##
#Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary), #Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary),
#Y P. Vojtechovsky (University of Denver, USA) #Y P. Vojtechovsky (University of Denver, USA)
@ -32,6 +32,7 @@ ReadPackage("loops", "data/small.tbl"); # small loops
ReadPackage("loops", "data/interesting.tbl"); # interesting loops ReadPackage("loops", "data/interesting.tbl"); # interesting loops
ReadPackage("loops", "data/nilpotent.tbl"); # nilpotent loops ReadPackage("loops", "data/nilpotent.tbl"); # nilpotent loops
ReadPackage("loops", "data/automorphic.tbl"); # automorphic loops ReadPackage("loops", "data/automorphic.tbl"); # automorphic loops
ReadPackage("loops", "data/rightbruck.tbl"); # right Bruck loops
# up to isotopism # up to isotopism
ReadPackage("loops", "data/itp_small.tbl"); # small loops up to isotopism ReadPackage("loops", "data/itp_small.tbl"); # small loops up to isotopism
@ -60,6 +61,7 @@ function( name )
elif name = "interesting" then return LOOPS_interesting_data; elif name = "interesting" then return LOOPS_interesting_data;
elif name = "nilpotent" then return LOOPS_nilpotent_data; elif name = "nilpotent" then return LOOPS_nilpotent_data;
elif name = "automorphic" then return LOOPS_automorphic_data; elif name = "automorphic" then return LOOPS_automorphic_data;
elif name = "right Bruck" then return LOOPS_right_bruck_data;
#up to isotopism #up to isotopism
elif name = "itp small" then return LOOPS_itp_small_data; elif name = "itp small" then return LOOPS_itp_small_data;
fi; fi;
@ -74,11 +76,8 @@ end);
InstallGlobalFunction( DisplayLibraryInfo, function( name ) InstallGlobalFunction( DisplayLibraryInfo, function( name )
local s, lib, k; local s, lib, k;
# up to isomorphism # up to isomorphism
if name = "left Bol" then if name = "left Bol" or name = "right Bol" then
s := "The library contains all nonassociative left Bol loops of order less than 17\nand all nonassociative left Bol loops of order p*q, where p>q>2 are primes."; s := Concatenation( "The library contains all nonassociative ", name, " loops of order less than 17\nand all nonassociative ", name, " loops of order p*q, where p>q>2 are primes." );
elif name = "right Bol" then
s := "The library contains all nonassociative right Bol loops of order less than 17\nand all nonassociative left Bol loops of order p*q, where p>q>2 are primes.";
name := "left Bol"; # using dual data
elif name = "Moufang" then elif name = "Moufang" then
s := "The library contains all nonassociative Moufang loops \nof order less than 65, and all nonassociative Moufang \nloops of order 81 and 243."; s := "The library contains all nonassociative Moufang loops \nof order less than 65, and all nonassociative Moufang \nloops of order 81 and 243.";
elif name = "Paige" then elif name = "Paige" then
@ -88,12 +87,9 @@ InstallGlobalFunction( DisplayLibraryInfo, function( name )
elif name = "Steiner" then elif name = "Steiner" then
s := "The library contains all nonassociative Steiner loops \nof order less or equal to 16. It also contains the \nassociative Steiner loops of order 4 and 8."; s := "The library contains all nonassociative Steiner loops \nof order less or equal to 16. It also contains the \nassociative Steiner loops of order 4 and 8.";
elif name = "CC" then elif name = "CC" then
s := "The library contains all nonassociative CC loops of order less than 28 \nand all nonassociative CC loops of order p^2 and 2*p for any odd prime p."; s := "The library contains all CC loops of order\n2<=2^k<=64, 3<=3^k<=81, 5<=5^k<=125, 7<=7^k<=343,\nall nonassociative CC loops of order less than 28,\nand all nonassociative CC loops of order p^2 and 2*p for any odd prime p.";
elif name = "RCC" then elif name = "RCC" or name = "LCC" then
s := "The library contains all nonassociative RCC loops of order less than 28."; s := Concatenation( "The library contains all nonassociative ", name, " loops of order less than 28." );
elif name = "LCC" then
s := "The library contains all nonassociative LCC loops of order less than 28.";
name := "RCC"; # using dual data
elif name = "small" then elif name = "small" then
s := "The library contains all nonassociative loops of order less than 7."; s := "The library contains all nonassociative loops of order less than 7.";
elif name = "interesting" then elif name = "interesting" then
@ -103,23 +99,27 @@ InstallGlobalFunction( DisplayLibraryInfo, function( name )
elif name = "automorphic" then elif name = "automorphic" then
s := "The library contains:\n"; s := "The library contains:\n";
s := Concatenation(s," - all nonassociative automorphic loops of order less than 16,\n"); s := Concatenation(s," - all nonassociative automorphic loops of order less than 16,\n");
s := Concatenation(s," - all commutative automorphic loops of order 3, 9, 27, 81,\n"); s := Concatenation(s," - all commutative automorphic loops of order 3, 9, 27, 81.");
s := Concatenation(s," - all commutative automorphic loops of order 243 that are central\n"); elif name = "left Bruck" or name = "right Bruck" then
s := Concatenation(s," extensions of Z_3 by F, where F is not the elem. ab. 3-group.\n"); s := Concatenation( "The library contains all ", name, " loops of orders 3, 9, 27 and 81." );
s := Concatenation(s,"Note: Abelian groups are included among the commutative loops.");
# up to isotopism # up to isotopism
elif name = "itp small" then elif name = "itp small" then
s := "The library contains all nonassociative loops of order less than 7 up to isotopism."; s := "The library contains all nonassociative loops of order less than 7 up to isotopism.";
else else
Info( InfoWarning, 1, Concatenation( Info( InfoWarning, 1, Concatenation(
"The admissible names for loop libraries are: \n", "The admissible names for loop libraries are: \n",
"[ \"left Bol\", \"right Bol\", \"Moufang\", \"Paige\", \"code\", \"Steiner\", \"CC\", \"RCC\", \"LCC\", \"small\", \"itp small\", \"interesting\", \"nilpotent\", \"automorphic\" ]." "\"automorphic\", \"CC\", \"code\", \"interesting\", \"itp small\", \"LCC\", \"left Bol\", \"left Bruck\", \"Moufang\", \"nilpotent\", \"Paige\", \"right Bol\", \"right Bruck\", \"RCC\", \"small\", \"Steiner\"."
) ); ) );
return fail; return fail;
fi; fi;
s := Concatenation( s, "\n------\nExtent of the library:" ); s := Concatenation( s, "\n------\nExtent of the library:" );
# renaming for data access
if name = "right Bol" then name := "left Bol"; fi;
if name = "LCC" then name := "RCC"; fi;
if name = "left Bruck" then name := "right Bruck"; fi;
lib := LOOPS_LibraryByName( name ); lib := LOOPS_LibraryByName( name );
for k in [1..Length( lib[ 1 ] ) ] do for k in [1..Length( lib[ 1 ] ) ] do
if lib[ 2 ][ k ] = 1 then if lib[ 2 ][ k ] = 1 then
@ -128,12 +128,12 @@ InstallGlobalFunction( DisplayLibraryInfo, function( name )
s := Concatenation( s, "\n ", String( lib[ 2 ][ k ] ), " loops of order ", String( lib[ 1 ][ k ] ) ); s := Concatenation( s, "\n ", String( lib[ 2 ][ k ] ), " loops of order ", String( lib[ 1 ][ k ] ) );
fi; fi;
od; od;
if name = "left Bol" or name = "right Bol" then if name = "left Bol" then
s := Concatenation( s, "\n (p-q)/2 loops of order p*q for primes p>q>2 such that q divides p-1"); s := Concatenation( s, "\n (p-q)/2 loops of order p*q for primes p>q>2 such that q divides p-1");
s := Concatenation( s, "\n (p-q+2)/2 loops of order p*q for primes p>q>2 such that q divides p+1" ); s := Concatenation( s, "\n (p-q+2)/2 loops of order p*q for primes p>q>2 such that q divides p+1" );
fi; fi;
if name = "CC" then if name = "CC" then
s := Concatenation( s, "\n 3 loops of order p^2 for every odd prime p,\n 1 loop of order 2*p for every odd prime p" ); s := Concatenation( s, "\n 3 loops of order p^2 for every prime p>7,\n 1 loop of order 2*p for every odd prime p" );
fi; fi;
s := Concatenation( s, "\n" ); s := Concatenation( s, "\n" );
Print( s ); Print( s );
@ -436,7 +436,48 @@ end);
InstallGlobalFunction( LOOPS_ActivateCCLoop, InstallGlobalFunction( LOOPS_ActivateCCLoop,
function( n, pos_n, m, case ) function( n, pos_n, m, case )
local T, x, y, k, a, b, p; local powers, p, i, k, F, basis, coords, coc, T, a, b, x, y;
powers := [ ,[4,8,16,32,64],[9,27,81],,[25,125],,[49,343]];
if n in Union( powers ) then # use cocycles
# determine p and position of n in database
p := Filtered([2,3,5,7], x -> n in powers[x])[1];
pos_n := Position( powers[p], n );
if not IsBound( LOOPS_cc_cocycles[p] ) then
# data not read yet, activate once
ReadPackage( "loops", Concatenation( "data/cc/cc_cocycles_", String(p), ".tbl" ) );
# decode cocycles and separate coordinates from a long string
for i in [1..Length(powers[p])] do
LOOPS_cc_cocycles[ p ][ i ] := List( LOOPS_cc_cocycles[ p ][ i ],
c -> LOOPS_DecodeCocycle( [ p^i, c[1], c[2] ], [0..p-1] )
);
LOOPS_cc_coordinates[ p ][ i ] := List( LOOPS_cc_coordinates[ p ][ i ],
c -> SplitString( c, " " )
);
od;
fi;
# data is now read
# determine position of loop in the database
k := 1;
while m > Length( LOOPS_cc_coordinates[ p ][ pos_n ][ k ] ) do
m := m - Length( LOOPS_cc_coordinates[ p ][ pos_n ][ k ] );
k := k + 1;
od;
# factor loop
F := CCLoop( n/p, LOOPS_cc_used_factors[ p ][ pos_n ][ k ] );
# basis
basis := List( LOOPS_cc_bases[ p ][ pos_n ][ k ],
i -> LOOPS_cc_cocycles[ p ][ pos_n ][ i ]
);
# coordinates
coords := LOOPS_cc_coordinates[ p ][ pos_n ][ k ][ m ];
coords := LOOPS_ConvertBase( coords, 91, p, Length( basis ) );
coords := List( coords, LOOPS_CharToDigit );
# cocycle
coc := (coords*basis) mod p;
coc := List( coc, i -> i+1 );
# return extension of Z_p by F using cocycle and trivial action
return LoopByExtension( CCLoop(p,1), F, List([1..n/p], i -> () ), coc );
fi;
if case=false then # use library of RCC loops, must recalculate pos_n if case=false then # use library of RCC loops, must recalculate pos_n
return LOOPS_ActivateRCCLoop( n, Position(LOOPS_rcc_data[ 1 ], n), LOOPS_cc_data[ 3 ][ pos_n ][ m ] ); return LOOPS_ActivateRCCLoop( n, Position(LOOPS_rcc_data[ 1 ], n), LOOPS_cc_data[ 3 ][ pos_n ][ m ] );
@ -543,39 +584,52 @@ end);
InstallGlobalFunction( LOOPS_ActivateAutomorphicLoop, InstallGlobalFunction( LOOPS_ActivateAutomorphicLoop,
function( n, m ) function( n, m )
local i, pos_n, factor_id, F, dim, coords, basis, coc; # returns the associated Gamma loop (which here always happens to be automorphic)
if IsEmpty( LOOPS_automorphic_cocycles ) then # only read on demand # improve later
ReadPackage( "loops", "data/automorphic/automorphic_cocycles.tbl"); local P, L, s, Ls, ct, i, j, pos, f;
# decode cocycles P := LeftBruckLoop( n, m );
for i in [1..3] do L := LeftMultiplicationGroup( P );;
LOOPS_automorphic_cocycles[ i ] := List( LOOPS_automorphic_cocycles[ i ], s := List(Elements(L), x -> x^2 );;
c -> LOOPS_DecodeCocycle( [ 3^(i+2), true, c ], [0,1,2] ) Ls := List([1..n], i -> LeftTranslation( P, Elements(P)[i] ) );;
); ct := List([1..n],i->0*[1..n]);;
od; for i in [1..n] do for j in [1..n] do
# separate coordinates (from a long string ) pos := Position( s, Ls[i]*Ls[j]*Ls[i]^(-1)*Ls[j]^(-1) );
for i in [1..3] do f := Elements(L)[pos];
LOOPS_automorphic_coordinates[ i ] := SplitString( LOOPS_automorphic_coordinates[ i ], " " ); ct[i][j] := 1^(f*Ls[j]*Ls[i]);
od; od; od;
fi; return LoopByCayleyTable(ct);
end);
#############################################################################
##
#F LOOPS_ActivateRightBruckLoop( n, m )
##
## Activates a right Bruck loop from the library.
InstallGlobalFunction( LOOPS_ActivateRightBruckLoop,
function( n, m )
local pos_n, factor_id, F, basis, coords, coc;
# factor loop # factor loop
pos_n := Position( [27,81,243], n ); pos_n := Position( [27,81], n );
factor_id := LOOPS_CharToDigit( LOOPS_automorphic_coordinates[ pos_n ][ m ][ 1 ] ); factor_id := LOOPS_CharToDigit( LOOPS_right_bruck_coordinates[ pos_n ][ m ][ 1 ] );
F := AutomorphicLoop( n/3, factor_id ); F := RightBruckLoop( n/3, factor_id );
# coordinates determining the cocycle # basis (only decode cocycles at first usage)
dim := Length( LOOPS_automorphic_bases[ pos_n ][ factor_id ] ); if IsString( LOOPS_right_bruck_cocycles[ pos_n ][ 1 ][ 3 ] ) then # not converted yet
coords := LOOPS_automorphic_coordinates[ pos_n ][ m ]; LOOPS_right_bruck_cocycles[ pos_n ] := List( LOOPS_right_bruck_cocycles[ pos_n ],
coords := coords{[2..Length(coords)]}; # remove the character that determines factor id coc -> LOOPS_DecodeCocycle( coc, [0,1,2] )
coords := LOOPS_ConvertBase( coords, 91, 3, dim );
coords := List( coords, LOOPS_CharToDigit );
# basis
basis := List( LOOPS_automorphic_bases[ pos_n ][ factor_id ],
i -> LOOPS_automorphic_cocycles[ pos_n ][ i ]
); );
fi;
basis := LOOPS_right_bruck_cocycles[ pos_n ];
# coordinates determining the cocycle
coords := LOOPS_right_bruck_coordinates[ pos_n ][ m ];
coords := coords{[2..Length(coords)]}; # remove the character that determines factor id
coords := LOOPS_ConvertBase( coords, 91, 3, Length( basis ) );
coords := List( coords, LOOPS_CharToDigit );
# calculate cocycle # calculate cocycle
coc := (coords*basis) mod 3; coc := (coords*basis) mod 3;
coc := List( coc, i -> i+1 ); coc := coc + 1;
# return extension of Z_3 by F using cocycle and trivial action # return extension of Z_3 by F using cocycle and trivial action
return LoopByExtension( AutomorphicLoop(3,1), F, List([1..n/3], i -> () ), coc ); return LoopByExtension( RightBruckLoop(3,1), F, List([1..n/3], i -> () ), coc );
end); end);
############################################################################# #############################################################################
@ -593,13 +647,7 @@ InstallGlobalFunction( LibraryLoop, function( name, n, m )
local lib, implemented_orders, NOL, loop, pos_n, p, q, divs, PG, m_inv, root, half, case, g, h; local lib, implemented_orders, NOL, loop, pos_n, p, q, divs, PG, m_inv, root, half, case, g, h;
# selecting data library # selecting data library
if name = "right Bol" then # using dual data
lib := LOOPS_LibraryByName( "left Bol" );
elif name = "LCC" then # using dual data
lib := LOOPS_LibraryByName( "RCC" );
else
lib := LOOPS_LibraryByName( name ); lib := LOOPS_LibraryByName( name );
fi;
# extent of the library # extent of the library
implemented_orders := lib[ 1 ]; implemented_orders := lib[ 1 ];
@ -614,7 +662,7 @@ InstallGlobalFunction( LibraryLoop, function( name, n, m )
# parameters for handling systematic cases, such as CCLoop( p^2, 1 ) # parameters for handling systematic cases, such as CCLoop( p^2, 1 )
pos_n := fail; pos_n := fail;
case := false; case := false;
if name="left Bol" or name="right Bol" then if name="left Bol" then
divs := DivisorsInt( n ); divs := DivisorsInt( n );
if Length( divs ) = 4 and not IsInt( divs[3]/divs[2] ) then # case n = p*q if Length( divs ) = 4 and not IsInt( divs[3]/divs[2] ) then # case n = p*q
q := divs[ 2 ]; q := divs[ 2 ];
@ -633,13 +681,13 @@ InstallGlobalFunction( LibraryLoop, function( name, n, m )
fi; fi;
if name="CC" then if name="CC" then
divs := DivisorsInt( n ); divs := DivisorsInt( n );
if Length( divs ) = 3 then # case p^2 if Length( divs ) = 3 and divs[ 2 ] > 7 then # case p^2, p>7
p := divs[ 2 ]; p := divs[ 2 ];
case := [p,"p^2"]; case := [p,"p^2"];
if not m in [1..3] then if not m in [1..3] then
Error("LOOPS: There are only 3 nonassociative CC-loops of order p^2 for an odd prime p."); Error("LOOPS: There are only 3 nonassociative CC-loops of order p^2 for an odd prime p.");
fi; fi;
elif Length( divs ) = 4 and not IsInt( divs[3]/divs[2] ) then # p*q elif Length( divs ) = 4 and not IsInt( divs[3]/divs[2] ) and not n=21 then # p*q
p := divs[ 3 ]; p := divs[ 3 ];
case := [p,"2*p"]; case := [p,"2*p"];
if not divs[2] = 2 then if not divs[2] = 2 then
@ -670,9 +718,6 @@ InstallGlobalFunction( LibraryLoop, function( name, n, m )
if name = "left Bol" then if name = "left Bol" then
loop := LOOPS_ActivateLeftBolLoop( pos_n, m, case ); loop := LOOPS_ActivateLeftBolLoop( pos_n, m, case );
SetIsLeftBolLoop( loop, true ); SetIsLeftBolLoop( loop, true );
elif name = "right Bol" then
loop := OppositeLoop( LOOPS_ActivateLeftBolLoop( pos_n, m, case ) );
SetIsRightBolLoop( loop, true );
elif name = "Moufang" then elif name = "Moufang" then
# renaming loops so that they agree with Goodaire's classification # renaming loops so that they agree with Goodaire's classification
PG := List([1..243], i->()); PG := List([1..243], i->());
@ -701,14 +746,15 @@ InstallGlobalFunction( LibraryLoop, function( name, n, m )
loop := LOOPS_ActivateSteinerLoop( n, pos_n, m ); loop := LOOPS_ActivateSteinerLoop( n, pos_n, m );
SetIsSteinerLoop( loop, true ); SetIsSteinerLoop( loop, true );
elif name = "CC" then elif name = "CC" then
if n in [2,3,5,7] then # use Cayley table for canonical cyclic group
loop := LoopByCayleyTable( LOOPS_DecodeCayleyTable( lib[ 3 ][ pos_n ][ m ] ) );
else
loop := LOOPS_ActivateCCLoop( n, pos_n, m, case ); loop := LOOPS_ActivateCCLoop( n, pos_n, m, case );
fi;
SetIsCCLoop( loop, true ); SetIsCCLoop( loop, true );
elif name = "RCC" then elif name = "RCC" then
loop := LOOPS_ActivateRCCLoop( n, pos_n, m ); loop := LOOPS_ActivateRCCLoop( n, pos_n, m );
SetIsRCCLoop( loop, true ); SetIsRCCLoop( loop, true );
elif name = "LCC" then
loop := OppositeLoop( LOOPS_ActivateRCCLoop( n, pos_n, m ) );
SetIsLCCLoop( loop, true );
elif name = "small" then elif name = "small" then
loop := LoopByCayleyTable( LOOPS_DecodeCayleyTable( lib[ 3 ][ pos_n ][ m ] ) ); loop := LoopByCayleyTable( LOOPS_DecodeCayleyTable( lib[ 3 ][ pos_n ][ m ] ) );
elif name = "interesting" then elif name = "interesting" then
@ -725,12 +771,19 @@ InstallGlobalFunction( LibraryLoop, function( name, n, m )
elif name = "nilpotent" then elif name = "nilpotent" then
loop := LOOPS_ActivateNilpotentLoop( lib[ 3 ][ pos_n ][ m ] ); loop := LOOPS_ActivateNilpotentLoop( lib[ 3 ][ pos_n ][ m ] );
elif name = "automorphic" then elif name = "automorphic" then
if not n in [27,81,243] then # use Cayley table if not n in [3, 9, 27, 81] then # use Cayley table
loop := LoopByCayleyTable( LOOPS_DecodeCayleyTable( lib[ 3 ][ pos_n ][ m ] ) ); loop := LoopByCayleyTable( LOOPS_DecodeCayleyTable( lib[ 3 ][ pos_n ][ m ] ) );
else # use cocycles else # use associated left Bruck loop
loop := LOOPS_ActivateAutomorphicLoop( n, m ); loop := LOOPS_ActivateAutomorphicLoop( n, m );
fi; fi;
SetIsAutomorphicLoop( loop, true ); SetIsAutomorphicLoop( loop, true );
elif name = "right Bruck" then
if not n in [27,81] then # use Cayley table
loop := LoopByCayleyTable( LOOPS_DecodeCayleyTable( lib[ 3 ][ pos_n ][ m ] ) );
else # use cocycles
loop := LOOPS_ActivateRightBruckLoop( n, m );
fi;
SetIsRightBruckLoop( loop, true );
# up to isotopism # up to isotopism
elif name = "itp small" then elif name = "itp small" then
return LibraryLoop( "small", n, lib[ 3 ][ pos_n ][ m ] ); return LibraryLoop( "small", n, lib[ 3 ][ pos_n ][ m ] );
@ -762,6 +815,8 @@ end);
#F InterestingLoop( n, m ) #F InterestingLoop( n, m )
#F NilpotentLoop( n, m ) #F NilpotentLoop( n, m )
#F AutomorphicLoop( n, m ) #F AutomorphicLoop( n, m )
#F LeftBruckLoop( n, m )
#F RightBruckLoop( n, m )
#F ItpSmallLoop( n, m ) #F ItpSmallLoop( n, m )
## ##
@ -770,7 +825,11 @@ InstallGlobalFunction( LeftBolLoop, function( n, m )
end); end);
InstallGlobalFunction( RightBolLoop, function( n, m ) InstallGlobalFunction( RightBolLoop, function( n, m )
return LibraryLoop( "right Bol", n, m ); local loop;
loop := Opposite( LeftBolLoop( n, m ) );
SetIsRightBolLoop( loop, true );
SetName( loop, Concatenation( "<right Bol loop ", String( n ), "/", String( m ), ">" ) );
return loop;
end); end);
InstallGlobalFunction( MoufangLoop, function( n, m ) InstallGlobalFunction( MoufangLoop, function( n, m )
@ -808,11 +867,15 @@ InstallGlobalFunction( RightConjugacyClosedLoop, function( n, m )
end); end);
InstallGlobalFunction( LCCLoop, function( n, m ) InstallGlobalFunction( LCCLoop, function( n, m )
return LibraryLoop( "LCC", n, m ); local loop;
loop := Opposite( RCCLoop( n, m ) );
SetIsLCCLoop( loop, true );
SetName( loop, Concatenation( "<LCC loop ", String( n ), "/", String( m ), ">" ) );
return loop;
end); end);
InstallGlobalFunction( LeftConjugacyClosedLoop, function( n, m ) InstallGlobalFunction( LeftConjugacyClosedLoop, function( n, m )
return LibraryLoop( "LCC", n, m ); return LCCLoop( n, m );
end); end);
InstallGlobalFunction( SmallLoop, function( n, m ) InstallGlobalFunction( SmallLoop, function( n, m )
@ -831,6 +894,18 @@ InstallGlobalFunction( AutomorphicLoop, function( n, m )
return LibraryLoop( "automorphic", n, m ); return LibraryLoop( "automorphic", n, m );
end); end);
InstallGlobalFunction( RightBruckLoop, function( n, m )
return LibraryLoop( "right Bruck", n, m );
end);
InstallGlobalFunction( LeftBruckLoop, function( n, m )
local loop;
loop := Opposite( RightBruckLoop( n, m ) );
SetIsLeftBruckLoop( loop, true );
SetName( loop, Concatenation( "<left Bruck loop ", String( n ), "/", String( m ), ">" ) );
return loop;
end);
InstallGlobalFunction( ItpSmallLoop, function( n, m ) InstallGlobalFunction( ItpSmallLoop, function( n, m )
return LibraryLoop( "itp small", n, m ); return LibraryLoop( "itp small", n, m );
end); end);

4
gap/iso.gd
View File

@ -2,7 +2,7 @@
## ##
#W iso.gd Isomorphisms and isotopisms [loops] #W iso.gd Isomorphisms and isotopisms [loops]
## ##
#H @(#)$Id: iso.gd, v 3.2.0 2015/06/12 gap Exp $ #H @(#)$Id: iso.gd, v 3.4.0 2016/12/13 gap Exp $
## ##
#Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary), #Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary),
#Y P. Vojtechovsky (University of Denver, USA) #Y P. Vojtechovsky (University of Denver, USA)
@ -23,6 +23,8 @@ DeclareOperation( "IsomorphismQuasigroups", [ IsQuasigroup, IsQuasigroup ] );
DeclareOperation( "IsomorphismLoops", [ IsLoop, IsLoop ] ); DeclareOperation( "IsomorphismLoops", [ IsLoop, IsLoop ] );
DeclareOperation( "QuasigroupsUpToIsomorphism", [ IsList ] ); DeclareOperation( "QuasigroupsUpToIsomorphism", [ IsList ] );
DeclareOperation( "LoopsUpToIsomorphism", [ IsList ] ); DeclareOperation( "LoopsUpToIsomorphism", [ IsList ] );
DeclareOperation( "QuasigroupIsomorph", [ IsQuasigroup, IsPerm ] );
DeclareOperation( "LoopIsomorph", [ IsLoop, IsPerm ] );
DeclareOperation( "IsomorphicCopyByPerm", [ IsQuasigroup, IsPerm ] ); DeclareOperation( "IsomorphicCopyByPerm", [ IsQuasigroup, IsPerm ] );
DeclareOperation( "IsomorphicCopyByNormalSubloop", [ IsLoop, IsLoop ] ); DeclareOperation( "IsomorphicCopyByNormalSubloop", [ IsLoop, IsLoop ] );

65
gap/iso.gi
View File

@ -2,7 +2,7 @@
## ##
#W iso.gi Isomorphisms and isotopisms [loops] #W iso.gi Isomorphisms and isotopisms [loops]
## ##
#H @(#)$Id: iso.gi, v 3.3.0 2016/10/26 gap Exp $ #H @(#)$Id: iso.gi, v 3.4.0 2017/08/24 gap Exp $
## ##
#Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary), #Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary),
#Y P. Vojtechovsky (University of Denver, USA) #Y P. Vojtechovsky (University of Denver, USA)
@ -466,30 +466,54 @@ end);
############################################################################# #############################################################################
## ##
#O IsomorphicCopyByPerm( Q, p ) #O QuasigroupIsomorph( Q, p )
## ##
## If <Q> is a quasigroup of order n and <p> a permutation of [1..n], returns ## If <Q> is a quasigroup of order n and <p> a permutation of [1..n], returns
## the quasigroup (Q,*) such that p(xy) = p(x)*p(y). ## the quasigroup (Q,*) such that p(xy) = p(x)*p(y).
## If <Q> is a loop, p is first composed with (1,1^p) to make sure
## that the neutral element of (Q,*) remains 1.
InstallMethod( IsomorphicCopyByPerm, "for a quasigroup and permutation", InstallMethod( QuasigroupIsomorph, "for a quasigroup and permutation",
[ IsQuasigroup, IsPerm ], [ IsQuasigroup, IsPerm ],
function( Q, p ) function( Q, p )
local ctQ, ct, inv_p; local ctQ, ct, inv_p;
ctQ := CanonicalCayleyTable( CayleyTable( Q ) ); ctQ := CanonicalCayleyTable( CayleyTable( Q ) );
# if Q is a loop and 1^p > 1, must normalize
if (IsLoop( Q ) and (not 1^p = 1)) then
p := p * (1, 1^p );
fi;
inv_p := Inverse( p ); inv_p := Inverse( p );
ct := List([1..Size(Q)], i-> List([1..Size(Q)], j -> ct := List([1..Size(Q)], i-> List([1..Size(Q)], j ->
( ctQ[ i^inv_p ][ j^inv_p ] )^p ( ctQ[ i^inv_p ][ j^inv_p ] )^p
) ); ) );
if IsLoop( Q ) then return LoopByCayleyTable( ct ); fi;
return QuasigroupByCayleyTable( ct ); return QuasigroupByCayleyTable( ct );
end); end);
#############################################################################
##
#O LoopIsomorph( Q, p )
##
## If <Q> is a loop of order n and <p> a permutation of [1..n] such that
## p(1)=1, returns the loop (Q,*) such that p(xy)=p(x)*p(y).
## If p(1)=c<>1, then the quasigroup (Q,*) is converted into loop
## via the isomorphism (1,c).
InstallMethod( LoopIsomorph, "for a loop and permutation",
[ IsLoop, IsPerm ],
function( Q, p )
return IntoLoop( QuasigroupIsomorph( Q, p ) );
end);
#############################################################################
##
#O IsomorphicCopyByPerm( Q, p )
##
## Calls LoopIsomorph( Q, p ) if <Q> is a loop,
## else QuasigroupIsotope( Q, p ).
InstallMethod( IsomorphicCopyByPerm, "for a quasigroup and permutation",
[ IsQuasigroup, IsPerm ],
function( Q, p )
if IsLoop( Q ) then
return LoopIsomorph( Q, p );
fi;
return QuasigroupIsomorph( Q, p );
end);
############################################################################# #############################################################################
## ##
#O IsomorphicCopyByNormalSubloop( L, S ) #O IsomorphicCopyByNormalSubloop( L, S )
@ -594,15 +618,13 @@ end);
## ##
## If L1, L2 are isotopic loops, returns true, else fail. ## If L1, L2 are isotopic loops, returns true, else fail.
# (MATH) First we calculate all principal loop isotopes of L1 of the form # (MATH) We check for isomorphism of L2 against all principal
# PrincipalLoopIsotope(L1, f, g), where f, g, are elements of L1. # isotopes of L1.
# Then we filter these up to isomorphism. If L2 is isotopic to L1, then
# L2 is isomorphic to one of these principal isotopes.
InstallMethod( IsotopismLoops, "for two loops", InstallMethod( IsotopismLoops, "for two loops",
[ IsLoop, IsLoop ], [ IsLoop, IsLoop ],
function( L1, L2 ) function( L1, L2 )
local istps, fg, f, g, L, phi, pos, alpha, beta, gamma, p; local f, g, L, phi, alpha, beta, gamma, p;
# make all loops canonical to be able to calculate isotopisms # make all loops canonical to be able to calculate isotopisms
if not L1 = Parent( L1 ) then L1 := LoopByCayleyTable( CayleyTable( L1 ) ); fi; if not L1 = Parent( L1 ) then L1 := LoopByCayleyTable( CayleyTable( L1 ) ); fi;
@ -619,20 +641,11 @@ function( L1, L2 )
if not Size(InnerMappingGroup(L1)) = Size(InnerMappingGroup(L2)) then return fail; fi; if not Size(InnerMappingGroup(L1)) = Size(InnerMappingGroup(L2)) then return fail; fi;
# now trying to construct an isotopism # now trying to construct an isotopism
istps := [];
fg := [];
for f in L1 do for g in L1 do for f in L1 do for g in L1 do
Add(istps, PrincipalLoopIsotope( L1, f, g )); L := PrincipalLoopIsotope( L1, f, g );
Add(fg, [ f, g ] );
od; od;
for L in LoopsUpToIsomorphism( istps ) do
phi := IsomorphismLoops( L, L2 ); phi := IsomorphismLoops( L, L2 );
if not phi = fail then if not phi = fail then
# must reconstruct the isotopism (alpha, beta, gamma) # must reconstruct the isotopism (alpha, beta, gamma)
# first figure out what f and g were
pos := Position( istps, L );
f := fg[ pos ][ 1 ];
g := fg[ pos ][ 2 ];
alpha := RightTranslation( L1, g ); alpha := RightTranslation( L1, g );
beta := LeftTranslation( L1, f ); beta := LeftTranslation( L1, f );
# we also applied an isomorphism (1,f*g) inside PrincipalLoopIsotope # we also applied an isomorphism (1,f*g) inside PrincipalLoopIsotope
@ -649,7 +662,7 @@ function( L1, L2 )
gamma := gamma * phi; gamma := gamma * phi;
return [ alpha, beta, gamma ]; return [ alpha, beta, gamma ];
fi; fi;
od; od; od;
return fail; return fail;
end); end);

13
gap/memory.gi
View File

@ -2,7 +2,7 @@
## ##
#W memory.gi Memory management [loops] #W memory.gi Memory management [loops]
## ##
#H @(#)$Id: memory.gi, v 3.3.0 2016/10/20 gap Exp $ #H @(#)$Id: memory.gi, v 3.4.0 2016/11/4 gap Exp $
## ##
#Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary), #Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary),
#Y P. Vojtechovsky (University of Denver, USA) #Y P. Vojtechovsky (University of Denver, USA)
@ -21,9 +21,14 @@ InstallGlobalFunction( LOOPS_FreeMemory, function( )
LOOPS_rcc_transitive_groups := []; LOOPS_rcc_transitive_groups := [];
LOOPS_rcc_sections := List( [1..Length(LOOPS_rcc_data[1])], i-> [] ); LOOPS_rcc_sections := List( [1..Length(LOOPS_rcc_data[1])], i-> [] );
LOOPS_rcc_conjugacy_classes := [ [], [] ]; LOOPS_rcc_conjugacy_classes := [ [], [] ];
# automorphic loops # cc loops
LOOPS_automorphic_cocycles := []; LOOPS_cc_used_factors := [];
LOOPS_automorphic_coordinates := []; LOOPS_cc_cocycles := [];
LOOPS_cc_bases := [];
LOOPS_cc_coordinates := [];
# right Bruck loops
LOOPS_right_bruck_cocycles := [];
LOOPS_right_bruck_coordinates := [];
GASMAN("collect"); GASMAN("collect");
return GasmanStatistics().full.deadkb; return GasmanStatistics().full.deadkb;
end); end);

6
gap/quasigroups.gd
View File

@ -2,7 +2,7 @@
## ##
#W quasigroups.gd Representing, creating and displaying quasigroups [loops] #W quasigroups.gd Representing, creating and displaying quasigroups [loops]
## ##
#H @(#)$Id: quasigroups.gd, v 3.2.0 2016/05/02 gap Exp $ #H @(#)$Id: quasigroups.gd, v 3.4.0 2017/10/17 gap Exp $
## ##
#Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary), #Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary),
#Y P. Vojtechovsky (University of Denver, USA) #Y P. Vojtechovsky (University of Denver, USA)
@ -24,10 +24,10 @@ DeclareRepresentation( "IsLoopElmRep",
IsPositionalObjectRep and IsMultiplicativeElementWithInverse, [1] ); IsPositionalObjectRep and IsMultiplicativeElementWithInverse, [1] );
## latin (auxiliary category for GAP to tell apart IsMagma and IsQuasigroup) ## latin (auxiliary category for GAP to tell apart IsMagma and IsQuasigroup)
DeclareCategory( "IsLatin", IsObject ); DeclareCategory( "IsLatinMagma", IsObject );
## quasigroup ## quasigroup
DeclareCategory( "IsQuasigroup", IsMagma and IsLatin ); DeclareCategory( "IsQuasigroup", IsMagma and IsLatinMagma );
## loop ## loop
DeclareCategory( "IsLoop", IsQuasigroup and IsMultiplicativeElementWithInverseCollection); DeclareCategory( "IsLoop", IsQuasigroup and IsMultiplicativeElementWithInverseCollection);

46
gap/quasigroups.gi
View File

@ -789,32 +789,34 @@ end );
InstallMethod( ViewObj, "for loop", InstallMethod( ViewObj, "for loop",
[ IsLoop ], [ IsLoop ],
function( L ) function( L )
if HasIsAssociative( L ) and IsAssociative( L ) then local PrintMe;
Print( "<associative loop of order ", Size( L ), ">");
elif HasIsExtraLoop( L ) and IsExtraLoop( L ) then PrintMe := function( name, L )
Print( "<extra loop of order ", Size( L ), ">"); Print( "<", name, " loop of order ", Size( L ), ">");
elif HasIsMoufangLoop( L ) and IsMoufangLoop( L ) then
Print( "<Moufang loop of order ", Size( L ), ">"); end;
elif HasIsCLoop( L ) and IsCLoop( L ) then if HasIsAssociative( L ) and IsAssociative( L ) then PrintMe( "associative", L );
Print( "<C loop of order ", Size( L ), ">"); elif HasIsExtraLoop( L ) and IsExtraLoop( L ) then PrintMe( "extra", L );
elif HasIsLeftBolLoop( L ) and IsLeftBolLoop( L ) then elif HasIsMoufangLoop( L ) and IsMoufangLoop( L ) then PrintMe( "Moufang", L );
Print( "<left Bol loop of order ", Size( L ), ">"); elif HasIsCLoop( L ) and IsCLoop( L ) then PrintMe( "C", L );
elif HasIsRightBolLoop( L ) and IsRightBolLoop( L ) then elif HasIsLeftBruckLoop( L ) and IsLeftBruckLoop( L ) then PrintMe( "left Bruck", L );
Print( "<right Bol loop of order ", Size( L ), ">"); elif HasIsRightBruckLoop( L ) and IsRightBruckLoop( L ) then PrintMe( "right Bruck", L );
elif HasIsLCLoop( L ) and IsLCLoop( L ) then elif HasIsLeftBolLoop( L ) and IsLeftBolLoop( L ) then PrintMe( "left Bol", L );
Print( "<LC loop of order ", Size( L ), ">"); elif HasIsRightBolLoop( L ) and IsRightBolLoop( L ) then PrintMe( "right Bol", L );
elif HasIsRCLoop( L ) and IsRCLoop( L ) then elif HasIsAutomorphicLoop( L ) and IsAutomorphicLoop( L ) then PrintMe( "automorphic", L );
Print( "<RC loop of order ", Size( L ), ">"); elif HasIsLeftAutomorphicLoop( L ) and IsLeftAutomorphicLoop( L ) then PrintMe( "left automorphic", L );
elif HasIsRightAutomorphicLoop( L ) and IsRightAutomorphicLoop( L ) then PrintMe( "right automorphic", L );
elif HasIsLCLoop( L ) and IsLCLoop( L ) then PrintMe( "LC", L );
elif HasIsRCLoop( L ) and IsRCLoop( L ) then PrintMe( "RC", L );
elif HasIsLeftAlternative( L ) and IsLeftAlternative( L ) then elif HasIsLeftAlternative( L ) and IsLeftAlternative( L ) then
if HasIsRightAlternative( L ) and IsRightAlternative( L ) then if HasIsRightAlternative( L ) and IsRightAlternative( L ) then
Print( "<alternative loop of order ", Size( L ), ">"); PrintMe("alternative", L );
else else
Print( "<left alternative loop of order ", Size( L ), ">"); PrintMe("left alternative", L );
fi; fi;
elif HasIsRightAlternative( L ) and IsRightAlternative( L ) then elif HasIsRightAlternative( L ) and IsRightAlternative( L ) then PrintMe( "right alternative", L );
Print( "<right alternative loop of order ", Size( L ), ">"); elif HasIsCommutative( L ) and IsCommutative( L ) then PrintMe( "commutative", L );
elif HasIsFlexible( L ) and IsFlexible( L ) then elif HasIsFlexible( L ) and IsFlexible( L ) then PrintMe( "flexible", L);
Print( "<flexible loop of order ", Size( L ), ">");
else else
# MORE ?? # MORE ??
Print( "<loop of order ", Size( L ), ">" ); Print( "<loop of order ", Size( L ), ">" );

2
tst/bol.tst
View File

@ -19,7 +19,7 @@ gap> IsomorphismLoops(B,LeftBolLoop(15,1));
gap> Q := RightBolLoop(15,1);; gap> Q := RightBolLoop(15,1);;
gap> AssociatedRightBruckLoop( Q ); gap> AssociatedRightBruckLoop( Q );
<right Bol loop of order 15> <right Bruck loop of order 15>
# TESTING EXACT GROUP FACTORIZATIONS # TESTING EXACT GROUP FACTORIZATIONS

2
tst/core_methods.tst
View File

@ -2,7 +2,7 @@
## ##
#W core_methods.tst Testing core methods G. P. Nagy / P. Vojtechovsky #W core_methods.tst Testing core methods G. P. Nagy / P. Vojtechovsky
## ##
#H @(#)$Id: core_methods.tst, v 3.3.0 2016/10/26 gap Exp $ #H @(#)$Id: core_methods.tst, v 3.4.0 2017/10/26 gap Exp $
## ##
#Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary), #Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary),
#Y P. Vojtechovsky (University of Denver, USA) #Y P. Vojtechovsky (University of Denver, USA)

7
tst/iso.tst
View File

@ -2,7 +2,7 @@
## ##
#W iso.tst Testing isomorphisms G. P. Nagy / P. Vojtechovsky #W iso.tst Testing isomorphisms G. P. Nagy / P. Vojtechovsky
## ##
#H @(#)$Id: iso.tst, v 3.2.0 2016/06/02 gap Exp $ #H @(#)$Id: iso.tst, v 3.4.0 2017/10/26 gap Exp $
## ##
#Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary), #Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary),
#Y P. Vojtechovsky (University of Denver, USA) #Y P. Vojtechovsky (University of Denver, USA)
@ -34,6 +34,11 @@ Group([ (1,2,3), (1,3,2) ])
gap> Q := DirectProduct( MoufangLoop( 32, 5 ) );; gap> Q := DirectProduct( MoufangLoop( 32, 5 ) );;
gap> Qp := IsomorphicCopyByPerm( Q, (2,3,4)(17,20) );; gap> Qp := IsomorphicCopyByPerm( Q, (2,3,4)(17,20) );;
gap> Qq := LoopIsomorph( Q, (2,3,4)(17,20) );;
gap> Qp = Qq;
false
gap> CayleyTable( Qp ) = CayleyTable( Qq );
true
gap> IsomorphismLoops( Q, Qp ); gap> IsomorphismLoops( Q, Qp );
(2,3,4)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32) (2,3,4)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)
gap> LoopsUpToIsomorphism( [Q,Qp] ); gap> LoopsUpToIsomorphism( [Q,Qp] );

54
tst/lib.tst
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@ -2,7 +2,7 @@
## ##
#W lib.tst Testing libraries of loops G. P. Nagy / P. Vojtechovsky #W lib.tst Testing libraries of loops G. P. Nagy / P. Vojtechovsky
## ##
#H @(#)$Id: lib.tst, v 3.3.0 2016/10/26 gap Exp $ #H @(#)$Id: lib.tst, v 3.4.0 2017/10/26 gap Exp $
## ##
#Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary), #Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary),
#Y P. Vojtechovsky (University of Denver, USA) #Y P. Vojtechovsky (University of Denver, USA)
@ -156,19 +156,34 @@ gap> LCCLoop(6,3); LCCLoop(25,119);
# CC LOOPS # CC LOOPS
gap> DisplayLibraryInfo("CC"); gap> DisplayLibraryInfo("CC");
The library contains all nonassociative CC loops of order less than 28 The library contains all CC loops of order
2<=2^k<=64, 3<=3^k<=81, 5<=5^k<=125, 7<=7^k<=343,
all nonassociative CC loops of order less than 28,
and all nonassociative CC loops of order p^2 and 2*p for any odd prime p. and all nonassociative CC loops of order p^2 and 2*p for any odd prime p.
------ ------
Extent of the library: Extent of the library:
2 loops of order 8 1 loop of order 2
1 loop of order 3
2 loops of order 4
1 loop of order 5
1 loop of order 7
7 loops of order 8
5 loops of order 9
3 loops of order 12 3 loops of order 12
28 loops of order 16 42 loops of order 16
7 loops of order 18 7 loops of order 18
3 loops of order 20 3 loops of order 20
1 loop of order 21 1 loop of order 21
14 loops of order 24 14 loops of order 24
55 loops of order 27 5 loops of order 25
3 loops of order p^2 for every odd prime p, 60 loops of order 27
437 loops of order 32
5 loops of order 49
14854 loops of order 64
5406 loops of order 81
84 loops of order 125
122 loops of order 343
3 loops of order p^2 for every prime p>7,
1 loop of order 2*p for every odd prime p 1 loop of order 2*p for every odd prime p
true true
@ -233,10 +248,7 @@ gap> CodeLoop( 64, 80 );
gap> DisplayLibraryInfo("automorphic"); gap> DisplayLibraryInfo("automorphic");
The library contains: The library contains:
- all nonassociative automorphic loops of order less than 16, - all nonassociative automorphic loops of order less than 16,
- all commutative automorphic loops of order 3, 9, 27, 81, - all commutative automorphic loops of order 3, 9, 27, 81.
- all commutative automorphic loops of order 243 that are central
extensions of Z_3 by F, where F is not the elem. ab. 3-group.
Note: Abelian groups are included among the commutative loops.
------ ------
Extent of the library: Extent of the library:
1 loop of order 3 1 loop of order 3
@ -249,7 +261,6 @@ Extent of the library:
2 loops of order 15 2 loops of order 15
7 loops of order 27 7 loops of order 27
72 loops of order 81 72 loops of order 81
118451 loops of order 243
true true
gap> AutomorphicLoop(15,2); gap> AutomorphicLoop(15,2);
@ -258,7 +269,24 @@ gap> AutomorphicLoop(15,2);
gap> AutomorphicLoop(27,1); gap> AutomorphicLoop(27,1);
<automorphic loop 27/1> <automorphic loop 27/1>
gap> AutomorphicLoop(243,100); gap> AutomorphicLoop(81,10);
<automorphic loop 243/100> <automorphic loop 81/10>
# RIGHT BRUCK LOOPS
gap> DisplayLibraryInfo("right Bruck");
The library contains all right Bruck loops of orders 3, 9, 27 and 81.
------
Extent of the library:
1 loop of order 3
2 loops of order 9
7 loops of order 27
72 loops of order 81
true
gap> RightBruckLoop(81,3);
<right Bruck loop 81/3>
gap> STOP_TEST( "lib.tst", 10000000 ); gap> STOP_TEST( "lib.tst", 10000000 );

12
tst/testall.g
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@ -2,15 +2,15 @@
## ##
#W testall.g Testing LOOPS G. P. Nagy / P. Vojtechovsky #W testall.g Testing LOOPS G. P. Nagy / P. Vojtechovsky
## ##
#H @(#)$Id: testall.g, v 3.0.0 2015/06/15 gap Exp $ #H @(#)$Id: testall.g, v 3.4.0 2017/10/26 gap Exp $
## ##
#Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary), #Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary),
#Y P. Vojtechovsky (University of Denver, USA) #Y P. Vojtechovsky (University of Denver, USA)
## ##
dirs := DirectoriesPackageLibrary( "loops", "tst" ); dirs := DirectoriesPackageLibrary( "loops", "tst" );
ReadTest( Filename( dirs, "core_methods.tst" ) ); Test( Filename( dirs, "core_methods.tst" ), rec( compareFunction := "uptowhitespace" ) );
ReadTest( Filename( dirs, "nilpot.tst" ) ); Test( Filename( dirs, "nilpot.tst" ), rec( compareFunction := "uptowhitespace" ) );
ReadTest( Filename( dirs, "iso.tst" ) ); Test( Filename( dirs, "iso.tst" ), rec( compareFunction := "uptowhitespace" ) );
ReadTest( Filename( dirs, "lib.tst" ) ); Test( Filename( dirs, "lib.tst" ), rec( compareFunction := "uptowhitespace" ) );
ReadTest( Filename( dirs, "bol.tst" ) ); Test( Filename( dirs, "bol.tst" ), rec( compareFunction := "uptowhitespace" ) );