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Merge branch 'Loops3.4.0'

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Incorporate the changes from 3.3.0 to 3.4.0 of LOOPS into this
development. These were mostly straightforward. The only conflicts were
in quasigroups.gd, in which all of the changes from this development
were selected, as "IsLatin" had already been removed.
This commit is contained in:
Glen Whitney 2017-10-30 00:33:44 -04:00
commit 91ba2744c1
63 changed files with 20298 additions and 29245 deletions
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8
PackageInfo.g
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@ -1,9 +1,9 @@
SetPackageInfo( rec(
PackageName := "loops",
Subtitle := "Computing with quasigroups and loops in GAP",
Version := "3.3.0",
Date := "26/10/2016",
ArchiveURL := "http://www.math.du.edu/loops/loops-3.3.0",
Version := "3.4.0",
Date := "27/10/2017",
ArchiveURL := "http://www.math.du.edu/loops/loops-3.4.0",
ArchiveFormats := "-win.zip .tar.gz",
Persons := [
@ -83,7 +83,7 @@ Dependencies := rec(
),
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
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
##
#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 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 := [
#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
[1,1,7,2,3,2,5,2,7,72,118451],
[1,1,7,2,3,2,5,2,7,72],
#the loops
[
#order 3 (Z_3)
#order 3 (Z_3, use left Bruck loops, placeholder only)
[
"201"
],
#order 6
[
@ -50,10 +32,8 @@ LOOPS_automorphic_data := [
"0325476301675421076455760132467102374523106543201",
"0325476310674520176545761023467013275432106452301"
],
#order 9 (two abelian groups)
#order 9 (two abelian groups, use left Bruck loops, placeholder only)
[
"204537861534867678012861207201345534",
"204537861534867678120862017012453345"
]
,
#order 10
@ -80,20 +60,13 @@ LOOPS_automorphic_data := [
"234068597BDAEC340189675DEBCA401297856ECDAB012375968CAEBD6897ADECB041328975DCABE430215689EABDC102439756CBDEA324107568BECAD21304BDEC0413258976DECA4302187569ABDE1024395687ECAB3241076895CABD2130469758",
"234067895BCDEA340178956CDEAB401289567DEABC012395678EABCD7968ADBEC012348579ECADB340129685DBECA123405796CADBE401236857BECAD23401DBEC0432156789ECAD3210478956ADBE1043295678BECA4321067895CADB2104389567"
],
#order 27 (commutative only, placeholder)
#order 27 (commutative only, use left Bruck loops, placeholder)
[
]
,
#order 81 (commutative only, placeholder)
[
]
,
#order 243 (commutative only, placeholder)
#order 81 (commutative only, use left Bruck loops, 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 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:
# If n = 2p or p^2, where p is a prime, then we call a method for
# cosntructing these loops.
# If n = 2p, where p is an odd prime, then we call an algebraic method for
# 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.
LOOPS_cc_data := [
#implemented orders
[ 8, 12, 16, 18, 20, 21, 24, 27],
#number of nonassociative loops of given order
[ 2, 3, 28, 7, 3, 1, 14, 55],
#the numbers of the loops in the RCC library
[ 2, 3, 4, 5, 7, 8, 9, 12, 16, 18, 20, 21, 24, 25, 27, 32, 49, 64, 81, 125, 343],
#number of loops of given order in the library
[ 1, 1, 2, 1, 1, 7, 5, 3, 42, 7, 3, 1, 14, 5, 60, 437, 5, 14854, 5406, 84, 122],
[
#order 8
[2,7],
#order 2 (Z_2)
["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
[53,73,89],
#order 16
[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 16 (placeholder only)
,
#order 18
[22,29,77,292,360,377,1133],
#order 20
@ -33,18 +54,12 @@ LOOPS_cc_data := [
#order 21
[104],
#order 24
[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]
[302,1025,2119,2182,2335,3066,4569,5176,5589,5997,7495,194830,225705,243216]
]
];
# The following can be used to point to CC loops of order 2p and p^2 in the library of RCC loops.
# order 6, [3]
# order 9, [5,4,3]
# order 10, [16]
# order 14, [97]
# order 22, [10346]
# order 25, [86,93,118]
# order 26, [151964]
LOOPS_cc_used_factors := [];
LOOPS_cc_cocycles := [];
LOOPS_cc_bases := [];
LOOPS_cc_coordinates := [];

1624
data/cc/cc_cocycles_2.tbl Normal file
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453
data/cc/cc_cocycles_3.tbl Normal file
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@ -0,0 +1,453 @@
#############################################################################
##
#W cc_cocycles_3.tbl G. P. Nagy / P. Vojtechovsky
##
#H @(#)$Id: cc_cocycles_3.tbl, v 3.4.0 2016/11/2 gap Exp $
##
#Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary),
#Y P. Vojtechovsky (University of Denver, USA)
##
# cocycles for CC loops of orders 9, 27, 81
LOOPS_cc_used_factors[3] :=
[
# factors of order 3
[1],
# factors of order 9
[1,2,4,5],
# factors of order 27
[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,55,56,57,58,60]
];
# end of used_factors
LOOPS_cc_cocycles[3] :=
[
# for factors of order 3
[
[false,"5"],
[false,"T"]
],
# for factors of order 9
[
[false,"Nb0XtD"],
[false,"1fMILI.euf?"],
[false,"A;R74)[>|%3D"],
[false,"YzxT&R77j7`jU"],
[false,"YzxT&R77(#k5t"],
[false,"8Jdd+{q'dYbah1x"],
[false,"~>hcUjmC(H{:6o#"],
[false,"dG.#{L(3kJl81+o4j"],
[true,"3#QhS<]EA"]
],
# for factors of order 27
[
[false,"hor~xnV^xMC5^'b9L&H<Q6<:}V9;#mn&98_0m=8A[H14j{yyKnG42|V4D"],
[false,"3&[N:Yq_~|RL'@B]s},@?,;Fq~B[I=:/@LdKq:nU3b.zc^f>FJroWT-;Qg*ugzyQ}6lZ^:7G1($3|i?Ut@6i3c^|R`M,n^N[d"],
[false,"7YpFoEn@TI$ke4t,QM]Xvq,.CVsRhg]7gu2XuPY3t[|Bf~$UMJ;#CQ5FfUpOEKpsQv7S5pXU.InvA;ZH5yZfw<-=(.[#?Y4B6%BAiCvwtr2MDV"],
[false,"f#y$l~9J2y[s%%JH0oTRODr]:>h~At1:wo}6niz#D:'#n-wM2V<vTnqvLy(NOKlk8_up6NzIavSp6RnpBq:V)#5Fg*i5+gt@Sn8Z$j`|RsP^yNnphD"],
[false,"929<Q3%|NK5~el_rNRioOfG@::OAICGt#Jq`Y}KwmbT7&aaAk$r|T&)lvOw:MoM:mWmkG|b$43Kj}UN6<+N]q9$}515#<OhPkRQy~Q7TIPKbqC))ute:2&"],
[false,"929<(pVnV`td-l5UUbV^aGx,t3/8g=0#CdP#6v@:>6p$N3lL{wx87,33DiqgK8T^]a&vAV~U~s3Z|7(%P=k>bBDOUWw4y1UvYBR'#a4<:G-L3/K@-#sS`Q"],
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[false,"7(>#S~Y335|CZBoN,?]q=&RECObN62xohkH6VM)O51&qZs7-#8Z6~^VH^tkEz:br>Q|$9rlSo4IHu8|mnnH_&}(3cUpMb/OCgVoBx+ZB&0v8*h&2a_wFu4y1)yc},B?A@+I]h8qH.l,bPeYT?T'nmJv|6Ci^v["],
[false,"7(>#S~Y6}APr]H||%#HA`x'}[]_cfa,ZThm`.fJL$62WOvs]s/>yz+9GeTU9`PyXWQdH~K(Eg&4nt9qa$&T((H15zi+V`o~~=,bX=dRrB1prZ6z{|^jBe1jD?Gd;}]||G{aU#w&@m^'S?4yp5Ccca66e8OUn2y"],
[false,"1M,aaVut'THt&#I8$eX`/zd6Y61SWR*J/0w}.GTnhbAq+k0VLwcc~&V+K*{_?-i/i`sfjE6qXjcoDcz)i3urwEngn[Ybj(oq%Mm%^zna/@?13MK/h{EQ6ryi|tGlIgn0N8Ntso]-D5z.yy}]'JDGum+l>brv_|S_BKO"],
[false,"AZDm2AHIgA;[|NfzI$dR(Dj^~S.sLOgyds,.>cn4wrBU@c:k]-g?eJ#hXXHGHoYF[dzO#~HKy9c8J0Uwa=ZTl0Of?_op*pZG/:wca8oWBajYfd:~F5u9jegk_||NXYC6pWK:i#pIbiM|l550M8()EG3SA?FX-@,t[^wp"],
[false,"1D2'6&,i(NNf~B^3D:i*Msb{*6lG#2TREWZ?q`[{=o'9P295`Xv&Dn]$kMK~bJ,{-1%6P:~kD~&<m?wa{&-M}3fU7D%$s3F>G>1=i1FPJ>|3]0Rgpj$H^N9-injHgbZWZA7]vUg(`_YA)gmI$;6<*l_G/Rm<yF@TlyeW~"],
[false,"36W4^H}F[+ubNCk8T&m?=Ddd`0KpjQsOyx+(o211g0ldXyrLqTYcEmv55b`%|BlBI);|SXLGj'#BNK/:/Ybb0K|3'*-pv(2x*9r(TN[w8sT6)r>.Yj,~7JAKn_J>y0LS^[e)o2]7gl3vmUXiFdl(udUiK[Mpl'lmP>)}^.?,g|?"],
[false,"36W4^H}vk^d:-(%z#7%Tve<CAop8bgx[sdKnvX$6)4Y:AnX)~/1YGkNrLM&uZ.XpI,q9s^S<,6(*,sdtB96P`m~qx&]za91{[j,kWug$:c:C/c6Zxc')AL#Z40k5qY5f}$4iW(_aD[slw.g`hT[+g8H>`V%/[}`6/Z#:khNb&0e"],
[true,"12.W./ADP*t:.tzo:SH>)sYG#t5nqzGr8ye0W<OmT^ev;19=H'^4;;|O~4CUgkUBNtwSscxvk=YP/uk$`A>nw="],
[true,"12.W./ADP*t:.tzo:SH>)sYG#t5nqzGr8ye0W<OmT^ev;19=L0/bwXp-IwNmkRRfx'qV<[)&PIU)t-Ed2~;Wil"],
[false,"36W4^H}vk^d:-(%=vMNy|5Z8tpnPC'5DCwGm$=*PK&63k_znE*V7fhM26AX^uSy7uhdU'2OE}R+GS9WZesF<;inFIE~Vxthf^nN&z{uImANt'dI=:MD,U|#WP{],};5X9>J?][ZhMgUv0+?t.`Wuck6gqW;)2{gy))^A%ll]Y}p"],
[false,"36W4^H}vk^d:-(&cg&]$|?~}8sh-?C&#dDX26wud{gA',cEWN.ai06sWB&;..LC.$?WG_;`+JDROHwnS')?U/,4^dab*a&G,2xbs41q$t[OtxyI-S6c%R3wxJP`K]pyH2}~R6:u)c>*ZQt.MW;:4G0T,z0ufpl%%LRBn9yfI46S"]
]
];
# end of cocycles
LOOPS_cc_bases[3] :=
[
# for factors of order 3
[
[1,2]
],
# for factors of order 9
[
[2,1,7,6,8,4,3],
[1],
[1,9],
[1,5]
],
# for factors of order 27
[
[2,3,1,26,14,39,27,33,28,5,41,42,40,43,8,4],
[2,1,26,14,32,5],
[2,1,24,11,31,36,9,4],
[2,1,25,13,30,4],
[2,1,26,14,5,4],
[2,1,26,14,5,4],
[2,1,21,33,5,4],
[2,1,22,4,32,5],
[2,1,18,5,31,4],
[2,1,19,6,30,4],
[2,1,26,14,4,5],
[2,1,26,14,7,4],
[2,1,14,33,5,4],
[2,1,4,14,32,5],
[2,1,4,14,32,5],
[2,1,5,11,31,4],
[2,1,6,13,30,4],
[2,1,7,12,29,4],
[2,1,26,14,5,4],
[2,1,26,14,4,5],
[2,1,26,14,4,5],
[2,1,26,14,5,4],
[2,1,26,14,6,4],
[2,1,26,14,7,4],
[2,1,26,14,5,4],
[2,1,26,14,4,5],
[2,1,26,14,4,5],
[2,1,26,14,5,4],
[2,1,26,14,7,4],
[2,1,26,14,6,4],
[2,1,14,33,5,4],
[2,1,15,4,32,5],
[2,1,16,5,31,4],
[2,1,17,6,30,4],
[2,1,26,14,5,4],
[2,1,26,14,4,5,45],
[2,1,26,14,5,4],
[2,1,26,14,6,4],
[2,1,26,14,5,4],
[2,1,26,14,4,5,38],
[2,1,26,14,5,4],
[2,1,26,14,6,4,37],
[2,1,21,33,5,4],
[2,1,22,4,32,5],
[2,1,23,4,32,5],
[2,1,18,5,31,4],
[2,1,19,6,30,4],
[2,1,20,7,29,4],
[2,1,21,33,5,4],
[44,10,34],
[1,46],
[1,47],
[1,48],
[1,35]
]
];
# end of bases
LOOPS_cc_coordinates[3] :=
[
# for factors of order 3
[
" 1 3 4 5"
],
# for factors of order 9
[
" 1 3 4 9 F R S U V b h @ [ ] ^ _ ` ~ 10 11 12 13 14 18 19 1A 1E 1F 1G 1H 1I 1\
K 1L 1Q 1R 1T 1U 1Z 1b 1f 1h 2| 2} 2~ 30 31 32 3O 3P 3Q 3R 3S 3T",
"1",
"1 4 5",
"1 4"
],
# for factors of order 27
[
" 1 3 4 9 F R S U V b h @ [ ] ^ _ ` ~ 10 11 12 13 14 18 19 1A 1E 1F 1G 1H 1I 1\
K 1L 1Q 1R 1T 1U 1Z 1b 1f 1h 2# 2% 2& 2' 2+ 2, 2- 2; 3p 3q 3r 3w 3y 3z 3# 3$ 3\
* 3- 3. 3/ 8A 8B 8S 8T 8U 8b 8c 8[ 8] 8^ 90 91 92 99 9A 9B 9I 9J 9R 9S 9a 9c A\
$ A, A. Br Bs Bz B$ B+ B, OC OD OE OI OK OL OM OO OP OQ OU OV OW OY OZ Od Oe O\
f Oj Ol Om On Oo Op Oq Ow Ox Oz O# O' O) O- O/ O: O; O= O> O^ O_ O` O| O} P2 P\
3 P4 P5 P7 PB PC PD PH PI PK PL PM PN PO PP PT PU PV PW PX PY Pc Pd Pe Pi Pj P\
k Pm Pn Pp Pq Pu Pw Px Pz P& P' P, P- P/ P: P; P@ P[ P] P^ P_ P` P| Q1 Q2 Q3 Q\
4 Q5 QB QC QH QI QJ QL QM QO QS QT QV QW Qb Qc Qd Qh Qi Qj Qk Ql Qm Qn Qo Qp Q\
t Qu Qv Qw Qx Qy Q& Q' Q) Q* Q. Q: Q; Q= Q] Q| Q} R1 R4 R9 RC RI Rt Ru Rz R# R\
$ R& R' R) R- R. R: R; SI SO SQ ST SZ Sc Sj Sm Sr Sx S# S& S- S. S: S; S? S[ S\
` S| S} S~ T1 T2 eE eK eW eX eY eZ ea eb ef eg eh el em en eo eq er et e` e{ e\
| e} e~ f0 f4 f5 f6 f7 f8 f9 fD fE fF fJ fK fL fM fN fP fQ fV fW fY fZ fe fg f\
k fm f; f< f= f[ f] f^ f_ f` f{ f| f} f~ g3 g4 g5 g6 g7 g8 gd ge gj gk gm gn g\
p gq gv gx gy g# g( g) g- g. g/ g: g< g= g> g? g^ g_ g` g| h9 hB hE hK hu hv h\
w hy hz h& h( h, h- h. h/ h: h< h= h> iJ iK iN iS iY iZ ib ie ig il in io it i\
u iw i% i+ i- i/ i: i; i< i= i[ i] i^ i_ i{ j0 j1 j6 j7 j8 -p -~ .1 .2 .3 .4 .\
5 .7 .8 .9 .A .B .C .E .F .I .J .L .M .N .O .P .Q .R .T .W .X .Z .c .e .f .h .\
i .j .l .m .p /+ /. /; /> /[ /_ :1 :4 :y :# :$ :& :' :) :* :+ :- :. :: :; :> :\
? :[ :] :_ :` :{ :} ;1 ;2 ;4 ;5 ;8 ;9 ;B ;C ;F ;G ;I ;J ;L ;Q ;R ;S ;T ;U ;V ;\
W ;X ;Y ;Z ;a ;b ;f ;g ;h ;l ;m ;n <e <h <k =U =V =W =X =Y =Z =d =e =f =j =k =\
l =p =q =r =s =t =u =v =y =$ =& =( =* =+ =. =; =< =? =] =} =~ >1 >2 >5 >6 >C >\
D >I |r }1 }3 }4 }5 }6 }A }B }C }D }E }K }L }P }Q }R }S }T }V }b }e }g }k }l }\
r ~- ~: ~= ~` 10# 10$ 10% 10& 10( 10) 10+ 10, 10- 10/ 10: 10< 10= 10> 10@ 10[ \
10] 10^ 10_ 10{ 10| 11S 11T 11U 11Y 11Z 11a 11h 11i 11j 12g 12m 13W 13X 13Y 13\
f 13g 13h 13r 13s 13t 13x 13# 13& 13( 13* 13, 13- 13: 13= 13> 13[ 13_ 18' 18) \
18- 18. 18| 18} 19Q 19W 19c 19f 19j 19m 1Ba 1Bc 1Bd 1Bg 1Bh 1Bk 1Bq 1Br 1Bw 1B\
# 1B: 1B[ 1B{ 1CC 1CJ 1F? 1G2 1G% 1G' 1G> 1G? 1G` 1G{ 1HU 1Ha 1Hk 1JY 1Ja 1Jh \
1Ji 1Ju 1Jv 1J* 1J. 1J? 2YR 2YS 2YU 2ZH 2ZI 2ZJ 2ZK 2ZL 2ZQ 2ZR 2ZT 2ZU 2ZV 2Z\
Z 2Za 2Zb 2Zf 2Zh 2gV 2gW 2hI 2hJ 2hO 2hP 2hR 2hS 2hU 2hV 2ha 2hc 2hd 2hf 2j> \
2j? 2j@ 2j] 2j^ 2j_ 2j` 2j{ 2j| 2k0 2k1 2k3 2k4 2k8 2kB 2q~ 2r@ 2r[ 2r] 2r^ 2r\
_ 2r` 2r{ 2r| 2r} 2r~ 2s0 2s1 2s2 2s3 2s4 2s6 2s7 2s8 2s9 2sA 2sB 2sD 2sE 2sF \
3F. 3F; 3GQ 3GS 3GW 3GY 3GZ 3Ga 3Gc 3Gd 3Gj 3Gk 3Gm 3Gn 3Gr 3Gt 3Gx 3Gz 3G# 3G\
$ 3G& 3G' 3G- 3G. 3G: 3G; 3G? 3G[ 3G` 3G| 3G} 3G~ 3H1 3H2 3H8 3H9 3HB 3HC 3IE \
3I{ 3I} 3I~ 3J1 3J2 3J4 3J5 3J6 3J8 3J9 3JB 3JC 3JF 3JG 3JI 3JJ 3JL 3JM 3JN 3J\
P 3JQ 3JS 3JT 3JV 3JW 3JX 3JZ 3Ja 3Jc 3Jd 3Jg 3Jh 3Jj 3Jk 3Jm 3Jn 3Jo 3Jq 3Ju \
3Jw 3J# 3J$ 3J& 3J' 3J* 3J+ 3J- 3J. 3K: 3Le 3Lk 3Ly 3L# 3L$ 3L& 3L' 3L( 3L* 3L\
+ 3L; 3L< 3L[ 3L] 3L^ 3L` 3L{ 3L~ 3M3 3M6 3M8 3M9 3MD 3MF 3MG 3MJ 3MK 3MM 3MP \
3MS 3MT 3MW 3MZ 3Ma 3Md 3Mg 3Mh 3Mj 3OR 3OT 3OU 3OW 3OX 3OZ 3Oa 3Ob 3Od 3Oe 3O\
g 3Oh 3Ok 3Ol 3On 3Oo 3Oq 3Or 3Os 3Ou 3Ov 3Ox 3Oy 3O# 3O$ 3O% 3O' 3O( 3O* 3O+ \
3O. 3O/ 3O; 3O< 3O> 3O? 3O@ 3O] 3O^ 3O` 3O{ 3O} 3O~ 3P0 3P2 3P3 3P5 3P6 3P9 3P\
A 3PC 3PD 3PF 3PG 3Q| 3Q~ 3R0 3R2 3R3 3R5 3R6 3R7 3R9 3RA 3RC 3RD 3RG 3RH 3RJ \
3RK 3RM 3RN 3RO 3RQ 3RR 3RT 3RU 3RW 3RX 3RY 3Ra 3Rb 3Rd 3Re 3Rh 3Ri 3Rk 3Rl 3R\
n 3Ro 3Rp 3Rr 3Rv 3Rx 3R$ 3R% 3R' 3R( 3R+ 3R, 3R. 3R/ 3Tz 3T$ 3T% 3T' 3T( 3T) \
3T+ 3T, 3T< 3T= 3T] 3T^ 3T_ 3T{ 3T| 3U0 3U2 3U4 3U6 3U7 3U9 3UA 3UD 3UE 3UG 3U\
H 3UK 3UL 3UN 3UO 3UQ 3UT 3UU 3UV 3UX 3Ua 3Ub 3Ue 3Uh 3Ui 3Uk 3Ul 3Z1 3Z3 3Z4 \
3Z6 3Z7 3Z8 3ZA 3ZB 3ZH 3ZI 3ZN 3ZO 3ZP 3ZR 3ZS 3ZU 3ZY 3ZZ 3Ze 3Zf 3Zl 3Zm 3Z\
o 3Zp 3Zq 3Zs 3Zw 3Zy 3Z% 3Z& 3Z( 3Z) 3Z, 3Z- 3Z/ 3Z: 3b# 3b% 3b& 3b( 3b) 3b* \
3b, 3b- 3b= 3b> 3b^ 3b_ 3b` 3b| 3b} 3c5 3cA 3cB 3cH 3cI 3cL 3cM 3cO 3cU 3cY 3c\
b 3cc 3cf 3ci 3cj 3h2 3h4 3h5 3h7 3h8 3h9 3hB 3hC 3hI 3hJ 3hO 3hP 3hQ 3hS 3hf \
3hg 3hm 3hn 3hr 3ht 3h) 3h* 3h: 3h; 3j' 3j. 3j? 3j{ 3j} 3j~ 3k1 3k5 3k6 3kB 3k\
C 3kI 3kJ 3kL 3kM 3kV 3kZ 3kg 3r% 3r, 3r{ 3r| 3s5 3s9 3sD 3sG 3sK 3sT 3sX 3sn \
3x1 3x3 3xD 3xE 3xK 3xL 3xV 3xX 3xb 3xc 3xr 3xs 3xw 3xy 3x% 3x& 3x? 3x@ 3z# 3z\
= 3z_ 3#1 3#3 3#4 3#6 3#7 3#8 3#A 3#B 3#H 3#I 3#N 3#O 3#R 3#e 3#l 8H_ 8I7 8IE \
8IJ 8IZ 8Ig 8P} 8Q2 8QI 8QN 8QU 8Qk AqJ AqZ Aqg A)R A)Y A)f LZ| La0 Lap Laq La\
s Lat Laz La# La% La& La* La, La: La< La= La> La@ La[ La| La} Lb0 Lb1 Lb5 Lb7 \
LbB LbD LbE LbF LbH LbI LbO LbP LbR LbS LbW LbY Lbc Lbe Lc_ Ld0 LdL LdM LdO Ld\
P LdS LdV LdW LdZ Ldb Ldc Ldd Ldg Ldi Ldj Ldl Ldm Ldn Ldq Lds Ldt Ldx Ldz Ld# \
Ld% Ld& Ld' Ld) Ld* Ld, Ld- Ld: Ld; Ld= Ld> Ld[ Ld_ Ld` Ld} Le0 Le1 Le2 Le4 Le\
5 Le7 Le8 Liq Lir Lit Liu Li# Li$ Li& Li' Li+ Li- Li; Li= Li> Li? Li[ Li] Li} \
Li~ Lj1 Lj2 Lj6 Lj8 LjC LjE LjF LjG LjI LjJ LjP LjQ LjS LjT LjX LjZ Ljd Ljf Ll\
M LlN LlP LlQ LlT LlW LlX Lla Llc Lld Lle Llh Llj Llk Llm Lln Llo Llr Llt Llu \
Lly Ll# Ll$ Ll& Ll' Ll( Ll* Ll+ Ll- Ll. Ll; Ll< Ll> Ll? Ll[ Ll] Ll` Ll{ Ll} Ll\
~ Lm1 Lm2 Lm3 Lm5 Lm6 Lm8 Lm9 LmB LmD LmF LmG LmJ LmM LmN LmQ LmT LmU LmX Lma \
Lmc Lme Lmg Lmh Lmj Lmk Lmn Lmo Lmq Lmr Lmu Lmv Lmx Lmy Lm$ Lm& Lm' Lm( Lm* Lm\
+ Lm. Lm; Lm< Lm? Lm[ Lm] Lm^ Lm{ Lm} Lm~ Ln1 Lyt Lyw Ly& Ly) Ly- Ly= Ly[ Ly^ \
Ly_ Lz0 Lz1 Lz4 Lz8 LzE LzG LzI LzL LzS LzV LzZ Lzf Lzj Lzp Lzt Lzw Lz# Lz& Lz\
- Lz= Lz[ Lz_ Lz} L#1 L#8 L#D L#E L#I L#K L#L L#P L#R L#S L$P L$S L$V L$Z L$c \
L$f L$g L$j L$m L$q L$t L$w L$# L$& L$) L$* L$- L$: L$> L$[ L$_ L$} L%1 L%4 L%\
5 L%8 L%B L%F L%I L%L L%P L%S L%V L%W L%Z L%c L%g L%j L%m L%q L%t L%w L%x L%# \
L%& L%* L%- L%: L%> L%[ L%_ L%` L%} L&1 L&` L&} L'0 L'1 L'5 L'7 L'8 L'B L'C L'\
E L'F L'I L'M L'P L'S L'W L'Z L'c L'd L'g L'j L'n L'q L't L'x L'# L'& L'' L'* \
L'- L'; L'> L'[ L'` L'} L(1 L(2 L(5 L(8 L(C L(F L(I L(M L(P L(S L(T L(W L(Z L(\
d L(g L(j L(n L(q L(t L(u L(x L(# L)u L)x L)' L)* L). L)> L)] L)` L*2 L*5 L*9 \
L*F L*J L*M L*T L*W L*a L*g L*k L*q L*u L*x L*$ L*' L*. L*> L*] L*` L*~ L+2 L+\
9 L+F L+J L+M L+Q L+T L.{ L.~ L/2 L/6 L/9 L/C L/D L/G L/J L/N L/Q L/T L/X L/a \
L/d L/e L/h L/k L/o L/r L/u L/y L/$ L/' L/( L/+ L/. L/< L/? L/] L/{ L/~ L:2 L:\
3 L:6 L:9 L:D L:G L:J L:N L:Q L:T L:U L:X L:a L:e L:h L:k L:o L:r L:u L:v L:y \
L:$ L;v L;y L;( L;+ L;/ L;? L;^ L;` L;{ L<2 L<3 L<6 L<A L<G L<I L<K L<N L<U L<\
X L<b L<h L<l L<r L<v L<y L<% L<( L</ L<? L<^ L<{ L=0 L=3 L=A L=G L=K L=N L=R \
L=U L>R L>U L>b L>h L>i L>l L>s L>v L>% L>+ L>, L>/ L>? L>@ L>^ L?0 L?5 L?6 L?\
7 L?A L?C L?H L?K L?R L?U L?Y L?e L?i L?l L?s L?v L?z L?( L?, L?/ L?@ L?^ L?| \
L@3 L@| L[0 L[3 L[7 L[A L[D L[E L[H L[K L[O L[R L[U L[Y L[b L[e L[f L[i L[l L[\
p L[s L[v L[z L[% L[( L[) L[, L[/ L[= L[@ L[^ L[| L]0 L]3 L]4 L]7 L]A L]E L]H \
L]K L]O L]R L]U L]V L]Y L]b L]f L]i L]l L]p L]s L]v L]w L]z L]% MHz MH# MH$ MH\
% MH& MH( MH) MH* MH+ MH, MH- MH/ MH: MH< MH= MH> MH@ MH[ MH^ MH_ MH` MH{ MH| \
MI0 MI2 MI3 MI5 MI8 MI9 MIB MIC MIE MIG MIH MIJ MIK MKV MKW MKX MKY MKZ MKc MK\
d MKe MKf MKg MKj MKl MKn MKp MKq MKs MKt MKu MKv MKw MKx MK$ MK% MK& MK* MK+ \
MK, MK- MK. MK/ MK: MK; MK< MK= MK> MK? MK_ MK{ MK} ML1 ML2 ML3 ML5 MLA MLC ML\
E MLI MLJ MN1 MN3 MN4 MN5 MN7 MN9 MNA MNB MNE MNH MNI MNL MNO MNP MNQ MNR MNS \
MNX MNZ MNd MNe MNh MNi MNk MNl MNm MNo MNu MNv MNw MN% MN& MN* MN; MN= MN? MX\
$ MX% MX& MX' MX( MX, MX- MX. MX/ MX; MX? MX@ MX` MX{ MX| MX} MX~ MY2 MY7 MYA \
MYB MYG MYI MYM MaX MaY MaZ Maa Mae Maf Mag Mah Mai Man Map Mar Mas Mau Maw Ma\
x May Maz Ma, Ma- Ma. Ma< Ma= Ma> Md3 Md5 Md7 Md9 MdB MdC MdD MdG MdK MdN MdR \
MdS MdT MdU Mdf Mdg Mdm Mdn Mf& Mf' Mf) Mf, Mf- Mf. Mf: Mf= Mf[ Mf_ Mf| Mf} Mg\
1 Mg4 Mg7 Mg8 MgB MgE MgI MgL MgO MgV MgY MgZ Mgc Mgj Mgp Mgt Mgw Mgz Mg# Mg& \
Mg) Mg- Mg: Mg= Mg[ Mg_ Mg` Mg| Mg} Mh1 Mh2 Mh4 Mh8 Mh9 MhB MhE MhI MhL MhP Mh\
V Mhc Mhf MiZ Mia Mic Mif Mig Mih Mij Mim Miq Mit Miw Mix Mi# Mi) Mi- Mi: Mi> \
Mi[ Mj1 Mj4 Mj5 Mj8 MjF MjL MjP MjV MjZ Mjc Mjg Mjj Mjq Mjt Mju Mjw Mjx Mj# Mj\
$ Mj& Mj* Mj+ Mj- Mj: Mj> Mj[ Mj` Mk1 Mk8 MkB Ml5 Ml6 Ml8 MlC MlD MlI MlP MlS \
MlT MlW Mlc Mlg Mlj Mln Mlq Ml# Ml& Ml' Ml* Ml; Ml[ Ml` Mm1 Mm5 Mm8 MmC MmF Mm\
P MmQ MmS MmT MmW MmX Mmd Mme Mmj Mmn Mmq Mmu Mm# Mm* Mm- Mn% Mn' Mn* Mn- Mn. \
Mn; Mn> Mn? Mn] Mn` Mn{ Mn} Mo2 Mo4 Mo5 Mo8 Mo9 MoB MoC MoF MoJ MoM MoO MoP Mo\
T Mog Mon Mou Mox Mo# Mo$ Mo' Mo* Mo. Mo; Mo> Mo] Mo^ Mo` Mo} Mo~ Mp0 Mp2 Mp5 \
Mp9 MpC MpF MpG MpP MpT Mpa MqY Mqa Mqd Mqh Mqn Mqo Mqr Mqu Mqv Mq' Mq+ Mq? Mq\
} Mq~ MrC MrE MrI MrJ MrT MrX Mrn Mrr Mrs Mru Mry Mrz Mr$ Mr+ Mr; Mr< Mr? Mr~ \
Ms6 Mt4 MtC MtG MtK MtN MtR Mta Mte Mtu Mt~ Mu3 MuJ MuO MuV Mul Muu Muy Mu( Mv\
, Mv. Mv< Mv= Mv^ Mv_ Mw3 Mw5 MwC MwG MwN MwP Mw$ Mw( Mw/ Mw[ Mw^ Mx6 Mx7 MxD \
MxE MxP MxW Mxa M$7 M$B M$C M$I M$J M$K M$N M$P M$R M$b M$f M$v M$| M%A M%H M%\
M M%c M%j M%v M%z M%) M)g M)i M)m M)n M)t M)u M*r M*t M*) M** M*: M*; M,C M,D \
M,E M,I M,J M,K M,O M,P M,Q M,} M-B M-I M-N M-d M-k M;b M;d M;q M;r M;x M;y M<\
1 M<H M<L M<v M<x M<$ M<% M<> M<? M_6 M_A M_Q M_a M_e M_l Qm` Qn2 Qn6 Qo6 QoM \
QoQ RM2 RM9 RMG RMN RMX RMk RO$ RO+ RO< RO^ RP3 RPG RU0 RU7 RUN RUO RUU RUb RU\
e RUi RUl RWz RW) RW^ RW_ RX0 RX7 RXA RXE RXH RXO RXV RXl T5s T5z T5& T6& T6@ \
T6_ T8m T8t T8) TDq TDx TD- TGk TG# TG' 2A-X 2A-Y 2A-Z 2A-a 2A-b 2A-c 2A-y 2A-\
z 2A-# 2A-$ 2A-% 2A-& 2A.7 2A.A 2A.F 2A.L 2A.N 2A.Q 2A.g 2A.m 2A.o 2A.r 2A.z 2\
A.% 2A.< 2A.] 2A/4 2A/7 2A/M 2A/P 2A/V 2A/Y 2A/e 2A/k 2A/n 2A/q 2A/( 2A/* 2A/+\
2A/- 2A:U 2A:W 2A:X 2A:Z 2A:$ 2A;C 2A;F 2A;K 2A;d 2A;g 2A;l 2A;o 2A;u 2A;# 2A\
;/ 2A;< 2A<J 2A<P 2A<b 2A<d 2A<h 2A<j 2A<% 2A<' 2A<+ 2A<- 2A=R 2A=U 2A=k 2A=n \
2A=s 2A=v 2A>9 2A>C 2A>T 2A>i 2A>l 2A>r 2A>, 2A>/ 2A|Z 2A|a 2A|b 2A|c 2A|d 2A|\
e 2A|# 2A|$ 2A|% 2A|& 2A|' 2A|( 2A}H 2A}N 2A}i 2A}o 2A}> 2A}_ 2A~g 2A~m 2A~* 2\
A~+ 2A~, 2A~: 2A~; 2A~< 2B0W 2B0X 2B0Y 2B0c 2B0d 2B0e 2B0x 2B1E 2B1H 2B1M 2B1f\
2B1i 2B1n 2B1q 2B1w 2B1; 2B1> 2B2L 2DFz 2DF( 2DF| 2DG3 2DI4 2DI7 2DIV 2DIY 2D\
N) 2DN} 2DPc 2DQ1 2V^0 2V^1 2V^2 2V^3 2V^4 2V^5 2V_S 2V_V 2V`} 2V{1 2V{4 2V{Z \
2V{c 2V{f 2V{h 2V{k 2V{= 2V{^ 2W14 2W15 2W16 2W2W 2W3~ 2W45 2W4a 2W4d 2W4g 2W4\
i 2W4o 2W4> 2W4[ 2W5F 2W5p 2W5y 2W5$ 2WH; 2WH[ 2WH^ 2WH{ 2WIV 2WIY 2WI& 2WI) 2\
WKR 2WKU 2WKX 2WL? 2WL] 2WME 2WNO 2WN~ 2WV[ 2WV| 2WW0 2WX` 2WYX 2Wa9 2WaT 2WbE\
2Wb[ 2a}' 2d%L 6r0f 6r0i 6r0l 6r31 6r34 6r9* 6r9: 6rB2 6rD9 6rDC 6rfC 6rfn A/\
5k A/kH",
"1 3 5 9 A C E R S U V X Z a b d e g i @ [ ^ ` ~ 10 12 14 18 19 1K 1L 1N 1P 1Q\
1R 1W 1Y 1Z 1a 1c 1d 2z 2$ 2( 2) 33 34 36 37 39 3B 3L 3M 3O 3P 3m 3o 4f 4h 4l\
4m",
"3 4 9 A B C D R U Y a b g @ [ ^ _ ` ~ 10 11 12 13 14 18 19 1E 1F 1G 1I 1K 1L \
1M 1N 1O 1P 1Q 1R 1S 1T 1U 1V 1Y 1e 1f 1g 1h 2z 2# 2$ 2% 2& 2' 2( 2+ 2, 2- 2. \
2< 2> 2] 2^ 2| 2} 2~ 30 31 32 36 37 3A 3E 3G 3I 3K 3L 3O 3P 3Q 3S 3T 3W 3Y 3Z \
3c 3d 3j 3k 3n 3p 3q 3s 3t 3y 3z 3# 3$ 3) 3= 3> 3@ 3] 3_ 3{ 3| 40 41 4E 4G 4I \
4O 4P 4Q 4T 8[ 8] 8^ 8{ 90 91 92 93 9F 9G 9H 9- 9. 9/ 9< 9@ 9[ 9] 9^ A5 A6 A7 \
A) A* A+ A/ A: A< A= A> B4 B5 B6 BB BC BD BJ BV BW Bb Bd Bf Bw Bx B% B( B) B* \
B` B{ CL C; C[ C~ DJ",
"3 9 A B C D R S T U V W X Y Z a c d e g h @ [ ] ^ _ ` ~ 10 11 12 13 14 18 19 \
1A 1K 1L 1M 1N 1P 1Q 1R 1S 1W 1X 1Y 1Z 1b 1c 1d 1e 2z 2# 2$ 2( 2) 2* 33 34 36 \
37 38 39 3A 3L 3M 3N 3O 3Q 3m 3n 4f 4g 4l 4n",
"9 A B C D E F G R S T U V W X Y Z a b c d e f g h i j k l m n o p q r ~ 10 11\
12 13 14 15 16 17 18 19 1A 1B 1C 1D 1E 1F 1G 1H 1I 1J 1K 1L 1M 1N 1O 1P 1Q 1R\
1S 1T 1V 1X 1Z 1a 1b 1c 1d 1e 1f 1g 1h 1i 1j 1k 1l 1m 1n 1o 1p 1q 1r 1s 1t 1u\
1v 1w 1x 1y 1z 1# 1$ 1% 1& 1' 1) 2+ 2, 2- 2. 2/ 2: 2; 2< 2= 2> 2? 2@ 2[ 2] 2^\
2_ 2` 2{ 2| 2} 2~ 30 31 32 33 34 35 3F 3G 3H 3I 3J 3N 3O 3P 3Q 3R 3S 3T 3U 3V\
3W 3X 3Y 3Z 3a 3b 3d 3y 3z 3# 3$ 3% 3& 3' 3( 3) 3* 3+ 3, 3- 3. 3/ 3: 3; 3< 3=\
3> 3? 3@ 3[ 3^ 4E 4F 4G 4H 4J 4L 4o 4q 4t 4u 4y",
"9 A B C D R S T U V W a b c d e f j k l m n o ~ 10 11 12 13 14 18 19 1A 1B 1C\
1D 1H 1I 1J 1K 1L 1M 1Q 1R 1S 1T 1U 1V 1Z 1a 1b 1c 1d 1e 1i 1j 1k 1o 1p 1q 1r\
1s 1t 1x 1y 1z 1# 1$ 1) 1* 2+ 2, 2- 2; 2< 2= 2> 2? 2@ 2_ 2` 2{ 2| 2} 2~ 30 31\
32 3F 3G 3H 3I 3J 3K 3O 3P 3Q 3R 3S 3T 3X 3Y 3a 3b 3y 3z 3# 3' 3( 3) 3* 3+ 3,\
3: 3; 3< 3= 3> 3@ 3[ 4E 4F 4G 4H 4I 4J 4u 4w",
"R S U V a b d e j l p r @ [ ] ^ _ ` ~ 10 11 15 16 17 1H 1J 1K 1M 1Q 1R 1T 1U \
1a 1b 1g 1h 2# 2) 2- 2: 2> 2[ 4f 4l 4p 4s 4) 4, 4= 4^ 4| 50 54 57 5E 5H 5O 5U"\
,
"R S U V X Z j k m n p r @ [ ^ ` ~ 10 12 14 18 19 1B 1D 1H 1I 1K 1L 1N 1P 1Q 1\
R 1T 1U 1W 1Y 1Z 1a 1c 1d 1f 1h 2z 2$ 2% 2& 2( 2) 2+ 2- 2. 2/ 2; 2< 2> 2@ 2[ 2\
] 2_ 2` 2| 2} 30 32 33 34 36 37 39 3B 3C 3D 3F 3G 3I 3K 3L 3M 3O 3P 3R 3S 3U 3\
W 3X 3Y 3a 3b 3d 3f 3g 3h 3j 3k 3m 3o 4f 4i 4j 4) 4* 4, 4. 4/ 4: 4` 4{ 4} 50 5\
1 52 54 55 57 58 5A 5C 5D 5E 5G 5H 5J 5L",
"R S U V X Z a b c d e f g h i @ [ ] ^ _ ` ~ 10 11 12 13 14 18 19 1A 1B 1C 1D \
1H 1J 1K 1L 1O 1P 1Q 1S 1T 1U 1X 1Y 1Z 1b 1c 1d 1g 1h 2# 2' 2( 2, 2: 2; 2? 2^ \
2_ 4) 4- 4; 54 58 5C",
"R T U V X Y a b c d e f g h i @ [ ] ^ _ ` ~ 10 11 12 13 14 18 19 1A 1B 1C 1D \
1H 1I 1J 1K 1L 1M 1N 1O 1P 1Q 1R 1S 1T 1U 1V 1W 1X 1Y 1Z 1a 1b 1c 1d 1e 1f 1g \
1h 2z 2# 2$ 2% 2& 2' 2( 2) 2* 2+ 2, 2- 2. 2/ 2: 2; 2< 2= 2> 2? 2@ 2[ 2] 2^ 2_ \
2` 2{ 2| 2} 2~ 30 31 32 33 34 35 36 37 38 39 3A 3B 3C 3D 3E 3F 3G 3H 3I 3J 3K \
3L 3M 3N 3O 3P 3Q 3R 3S 3T 3U 3V 3W 3X 3Y 3Z 3a 3b 3c 3d 3e 3f 3g 3h 3i 3j 3k \
3l 3m 3n 3o 4) 4* 4+ 4, 4- 4. 4/ 4: 4; 4` 4{ 4} 50 51 52 54 55 56 57 58 59 5A \
5B 5C 5M 5O 5P 5Q 5S 5T",
"R S T U V W X Y Z a b c d e f g h i j k l m n o p q r @ [ ] ^ _ ` { | } ~ 10 \
11 12 13 14 15 16 17 18 19 1A 1B 1C 1D 1E 1F 1G 1H 1I 1J 1N 1O 1P 1Q 1R 1S 1W \
1X 1Y 1Z 1a 1b 1f 1g 1h 1i 1j 1k 1o 1p 1q 1r 1s 1t 1x 1y 1z 1# 1$ 1% 1) 1* 1+ \
2% 2& 2' 2. 2/ 2: 2[ 2] 2^ 3v 3w 3x 3' 3( 3) 3: 3; 3< 4) 4* 4+ 54 55 56",
"R S T U V W a b c d e f j k l m n o @ [ ] ^ _ ` ~ 10 11 12 13 14 18 19 1A 1B \
1C 1D 1H 1J 1K 1L 1Q 1S 1T 1U 1Z 1b 1c 1d 1i 1k 1p 1q 1r 1t 1y 1z 1# 1% 1* 1+ \
2# 2( 2, 2; 2? 2_ 3r 3w 3# 3( 3, 3; 4) 4- 54 58",
"@ [ ] ^ _ ` { | } ~ 10 11 12 13 14 15 16 17 18 19 1A 1B 1C 1D 1E 1F 1G 1H 1J \
1M 1N 1P 1Q 1R 1T 1U 1W 1X 1a 1b 1d 1e 1g 1h 1i 1k 1l 1n 1o 1q 1r 1s 1u 1v 1x \
1y 1$ 1% 1' 1( 1* 1+ 2# 2& 2) 2} 31 34 3P 3S 3V 3r 3u 3x 3y 3$ 3' 3+ 3. 3;",
"@ [ ] ^ _ ` { | } ~ 10 11 12 13 14 15 16 17 18 19 1A 1B 1C 1D 1E 1F 1G 1H 1I \
1J 1K 1L 1M 1N 1O 1P 1S 1T 1U 1V 1W 1X 1Y 1Z 1a 1b 1c 1d 1e 1f 1g 1h 1j 1l 1m \
1n 1o 1p 1q 1r 1s 1t 1u 1v 1w 1x 1y 1z 1# 1$ 1% 1& 1' 1( 1) 1* 1+ 2$ 2% 2& 2' \
2( 2) 2+ 2, 2- 2. 2/ 2: 2; 2< 2= 2> 2? 2@ 2[ 2] 2^ 2_ 2` 2{ 2| 2} 2~ 30 31 32 \
33 34 35 36 37 38 39 3A 3B 3C 3D 3E 3G 3I 3L 3M 3N 3O 3P 3Q 3R 3S 3T 3U 3V 3W \
3X 3Y 3Z 3b 3d 3e 3f 3g 3h 3i 3j 3k 3l 3m 3n 3o 3r 3u 3v 3w 3x 3y 3z 3# 3$ 3% \
3& 3' 3( 3) 3+ 3- 3. 3/ 3: 3; 3< 3= 3> 3? 3@ 3[ 3] 3^ 3_ 3{ 3| 3} 3~ 40 41 42 \
43 44 46 48 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 4R 4T 4U 4V 4W 4X \
4Y 4Z 4a 4b 4c 4d 4e",
"[ ^ _ ` { | } ~ 10 11 12 13 14 15 16 17 18 19 1A 1B 1C 1D 1E 1F 1G 1I 1K 1N 1\
O 1P 1S 1T 1U 1V 1W 1X 1Y 1Z 1a 1b 1c 1d 1e 1g 1i 1j 1k 1l 1m 1n 1o 1q 1t 1u 1\
v 1x 1# 1$ 1% 1& 1' 1( 1) 1* 1+ 2$ 2% 2& 2( 2+ 2, 2- 2. 2/ 2: 2; 2< 2= 2? 2^ 2\
_ 2` 2{ 2| 2} 2~ 30 31 32 33 34 35 36 37 38 39 3A 3B 3C 3D 3E 3F 3G 3H 3J 3L 3\
N 3O 3P 3Q 3R 3S 3T 3U 3V 3W 3X 3Y 3Z 3b 3d 3e 3f 3h 3j 3r 3s 3t 3u 3y 3z 3# 3\
$ 3% 3& 3' 3( 3) 3+ 3- 3. 3/ 3< 3= 3> 3? 3@ 3[ 3] 3^ 3_ 3` 3{ 3| 3} 3~ 40 41 4\
2 43 44 46 49 4B 4C 4D 4E 4F 4G 4J 4K 4L 4M 4N 4O 4P 4R 4T 4U 4V 4X 4Z 4e",
"@ [ ] ^ _ ` { | } 15 18 1A 1B 1C 1F 1G 1K 1O 1Q 1S 1T 1U 1X 1Y 1# 1% 1& 1' 1*\
1+ 2# 2, 37 3G 3P 3r 3s 3# 3$ 3(",
"@ [ ^ _ ` { | } 12 17 18 19 1A 1B 1C 1E 1G 1L 1M 1O 1Q 1S 1T 1U 1V 1W 1X 1Y 1\
# 1$ 1% 1& 1' 1( 1) 1* 1+ 2# 2$ 2% 2& 2( 2* 2+ 2, 2- 2. 2/ 2; 2< 2= 2[ 2] 2{ 3\
6 37 38 39 3B 3C 3D 3E 3F 3G 3I 3J 3K 3L 3M 3N 3O 3Q 3R 3S 3T 3U 3V 3W 3q 3r 3\
s 3t 3v 3y 3z 3# 3$ 3% 3' 3( 3) 3- 3. 3: 3{ 3| 3} 3~ 41 42 43 44 45 48 49 4A 4\
B 4C 4D 4E 4G 4I 4J 4K 4L 4M",
"] ^ ` { | } 12 15 17 18 19 1A 1B 1C 1E 1F 1K 1L 1O 1Q 1T 1U 1V 1W 1X 1# 1$ 1%\
1& 1' 1( 1) 1+ 2# 2$ 2& 2' 2) 2* 2+ 2, 2- 2. 2: 2; 2< 2= 2` 36 37 38 39 3A 3C\
3D 3E 3F 3L 3N 3O 3P 3R 3S 3T 3U 3V 3q 3r 3t 3w 3x 3y 3z 3# 3$ 3& 3' 3( 3) 3.\
3< 3{ 3| 3} 3~ 40 42 43 44 4A 4B 4E 4F 4H 4J 4K 4L",
"@ ] ^ ` ~ 10 12 13 18 19 1A 1B 1C 1D 1Q 1T 1a 1b 1d 1e 1j 1p 1# 1$ 1) 1* 2# 2\
$ 2) 2* 2+ 2, 2- 2; 2< 2= 2| 2} 30 31 36 37 38 39 3A 3B 3F 3H 3I 3K 3O 3Q 3T 3\
Y 3Z 3b 3c 3q 3w 3y 3z 3# 3' 3( 3) 3+ 3; 3= 3> 3@ 3[ 3{ 3| 3} 3~ 40 41 45 47 4\
8 4A 4E 4G 4H 4J 4O 4P 4R",
"^ _ { | 15 16 18 19 1E 1F 1Z 1a 1f 1o 1q 1# 1% 1) 2% 2' 2. 2: 2[ 2^ 30 31 39 \
3A 3I 3J 3R 3a 3b 3v 3x 3^ 3_",
"@ [ ^ _ ~ 10 15 16 18 19 1B 1C 1E 1F 1T 1Z 1c 1d 1i 1k 1& 1) 1+ 2% 2( 2+ 2- 2\
. 2: 2; 2= 2_ 2{ 2| 2} 33 34 36 37 39 3A 3C 3D 3F 3G 3I 3J 3O 3R 3X 3Y 3d 3e 3\
s 3u 3v 3y 3$ 3& 3' 3* 3, 3= 3> 3^ 3_ 3{ 3| 40 42 43 45 46 48 49 4H 4N 4T",
"@ ] ^ ` { } ~ 10 12 13 15 16 18 19 1A 1B 1C 1D 1E 1F 1G 1Q 1T 1W 1b 1d 1e 1h \
1j 1m 1p 1# 1$ 1' 1) 1* 2* 2+ 2, 2- 2. 2/ 2: 2; 2< 2= 2| 31 33 34 36 37 38 39 \
3A 3B 3C 3D 3E 3F 3H 3I 3K 3L 3N 3Z 3b 3c 3e 3f 3q 3t 3w 3$ 3% 3& 3' 3( 3) 3+ \
3; 3= 3> 3@ 3^ 3_ 3{ 3| 3} 3~ 40 41 43 44 45 48 4A 4B 4D 4G 4H 4J 4K 4M 4O",
"[ ] ^ _ { } 11 13 15 18 1A 1C 1D 1E 1F 1e 1g 1k 1m 1o 1# 1( 1* 2, 2? 37 3G 3Y\
3r 3t 3? 3^",
"@ [ _ ` { } ~ 11 12 13 16 17 18 19 1A 1B 1C 1D 1E 1F 1G 1a 1e 1f 1i 1m 1q 1% \
2+ 2, 2- 2. 2/ 2: 2; 2< 2= 2@ 2[ 2` 2| 2~ 31 36 37 38 39 3A 3B 3C 3D 3E 3F 3G \
3J 3K 3L 3N 3X 3a 3b 3e 3f 3q 3u 3v 3& 3' 3* 3< 3= 3? 3@ 3_ 3` 3{ 3} 3~ 40 43 \
44 45 46 49 4A 4B 4D",
"[ _ | ~ 12 15 18 1A 1B 1D 1E 1G 1a 1d 1g 1i 1k 1l 1n 1o 1q 2+ 2, 2- 2. 2/ 2: \
2; 2< 2= 2@ 2^ 2{ 38 3X 3Y 3Z 3a 3b 3c 3d 3e 3f 3# 3% 3& 3) 3* 3- 3: 3` 4E 4H \
4K 4N 4Q 4T",
"@ [ ] 18 19 1A 1D 1Q 1R 1S 1f 1g 1i 1j 1k 1l 1n 1& 2+ 2, 2- 2. 2/ 2: 2< 2> 2?\
2@ 2[ 2] 2^ 37 3M 3X 3Y 3Z 3d 3f 3s 3t 3u 3z 3' 3( 3) 3= 3> 3? 4P",
"{ | } 15 16 17 1D 1E 1G 1b 1q 1$ 2/",
"[ ~ 12 18 1A 1d 1i 1o 1q 2; 2< 2= 2@ 2{ 38 3Z 3a 3b 3c 3) 3* 3: 4H 4N 4Q",
"@ 18 1U 1c 1i 1o 1q 2; 2= 2> 2? 2_ 2{ 3X 3a 3b 3v 3( 3=",
"] 11 13 1A",
"@ [ ^ _ ~ 12 15 1H 1K",
"^ _ ` { | 15 16 17 18 19 1A 1B 1C 1D 2| 2} 2~ 3g 3h 3i 3{ 3| 3} 4B 4C 4D 4W 4\
X 4Y",
"@ [ ^ 10 12 1H 1Z",
"] _ ` | } 10 12 16 38 3Q 3Y 3> 3^ 3{ 44 4G",
"[ ] _ ` 11 14 1Q 1T 1# 1) 2~ 32 48",
"1 4 A D S V b e k n [ ] _ ` 10 11 13 14 19 1A 1C 1D 1I 1J 1L 1M 1R 1S 1U 1V 2\
+ 2. 2[ 2| 30 36 39 3X 3a 3{ 8B 8C 8H 8I 91 92 94 95 9[ 9] 9_ 9`",
"] 11 14 17 1Q 1T 1& 1) 32",
"~ 10 14 19",
"2? 2] 2` 3y 3$ 3'",
"1 A B S T b c k l 10 19 1R 1a 1s 1$ 2# 2$ 2, 2- 2> 2? 2@ 2[ 2} 2~ 37 38 3G 3H\
3P 3Q 3Y 3Z 3g 3h 3i 3j 3q 3z 3+ 3> 3| 46 4F 4O 4X 4h 4z 4+ 4| 56 5O 8B 8C 8d\
8' 91 9[ DX DY Dy",
"2` 3'",
"1 4 S V Y [ _ | 1I 1L 2# 2& 2( 2) 2} 31 34 3P 3S 3U 3V 3a 3q 3w 3> 3_ 4F 4L 8\
5 86 8X DX DY EN",
"2| 30 54",
"3m 3o 4} 50",
"4] 4^ 5P 5Q",
"3F",
"3U 4-",
"59 5Q",
"54 57",
"1 3 4 9 A B D E F",
"1 4 5",
"1",
"1",
"1 4 5"
]
];
# end of coordinates

74
data/cc/cc_cocycles_5.tbl Normal file
View File

@ -0,0 +1,74 @@
#############################################################################
##
#W cc_cocycles_5.tbl G. P. Nagy / P. Vojtechovsky
##
#H @(#)$Id: cc_cocycles_5.tbl, v 3.4.0 2016/11/2 gap Exp $
##
#Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary),
#Y P. Vojtechovsky (University of Denver, USA)
##
# cocycles for CC loops of orders 25, 125
LOOPS_cc_used_factors[5] :=
[
# factors of order 5
[1],
# factors of order 25
[1,2,3]
];
# end of used_factors
LOOPS_cc_cocycles[5] :=
[
# for factos of order 5
[
[false,"Rw/*u"],
[true,"1}Q"]
],
# for factors of order 25
[
[false,"OB:_wPw=gcg28[}yWf2ey+Z^a|z'1V'7>:(U12L]NduxhqC9%,^lAJ$lGd+R&xY-MqDpp$|#k&O4;LJP,/;?HXv|,rf/D$@QWIYw)00]/z12+A^-T4'a;q[6kygZiInW<D6*"],
[false,"5zn,d/|kkGgr%mcn_WHG3*0,ASnvRTxKtN3H{sAfeZWm+hXd4Pir?wn?i&ofWgj9;Xc)S1(~da)Seu)&-N0}O)2(Ba@O})-wc~I=^[B?bo]7Dd=`Ewx,fcQ])TH}(3Ja-1v@]*pk:GWpgRlBlS:_GcQ{e%Bb}Q[A#o%Npw~B"],
[true,"3UiB~MDkk697Yh?ShcYJeyp/|.`D@M+V9qeR^]C/aUU_noye{VyHW{?v:X6@e:z#_ekgzg"],
[false,"X%yesf}wK3UZMQ*xIK.GD?Sxv(}[jQ2q$FS|ieoo1k-y3WDsv+XBZI<su/=f9#Mmyi6~'JYd:HVgUh?sO[^q1.{>[@Nn&c%0Q-FD}lch1U%7@Oo[j_R8yst}2j{WKEdidN}Yl&|3eGR+:84/Nabq]`X&QKwK&vZ-sie?M/BC>Tmrc^/a$T"],
[false,"X%yesf}wK3UZMQ*xIK.GD?Sxv(}[jQ2q$FS|ieoo1k-y3WDtKij(unBRDF=K]Z@_|wuO):aV9yhoF2b:5Bs97*L<}5FmM%dw;'|6T@_7H[$Ct8?-CP,n.X}Nn5/:2G_%LcBfYvg%:0q]<{qmQ~/<i9CAE9ftlgD}Oqm#c5]BoTXcSK<%M&"],
[false,"1UkuTqX=UnGoF4NH%>croy[i,GD]ZRNsMLN|KV{vh*9Lwv-)6:Cc,`6iV59@|1U93Tq^l)+1ijGcYs=z_YHH+e:@|_H9U'#7}QY4AR?&5)*v>YjLVkgi0k<=2SCh`CF-K)+x1Gs@#$Tb&XLToQ?iFyh%~EbvIr)I.W-URMAN*ZNfNc8cfW$G/acm=*tinG0@'4{Wx?1:30.E"],
[false,"BdI9~-PG:lXzbHWaF^p8SC$6zN:}=6PxaofDoilEd$w~oOR]>|xP@e'^89q2{uqI6Kw5>vT>-/UMS+d_+?aHbF(#4]O5{+:'LcAL*|N(L`eC`1BY-V+Z_%YQ'7N[w(*}/L1tP|k1,m9DP51]@JgMu,1fH5G.zH%w>(TD9Q<kpxZ+kD2@4wz~9Y:n(5+Tc@BIyC-$xGrTL}y%}"],
[true,"F$)1]6{~(SKUfTbwmVWma(rR`jHYTl,=Sh,}Nr58.oA1y6;4J%5tB7jn[gLrY2|L-?v73j:S^LUfY'g(447])_x-;1//p^O$%iT)$9wEtL"],
[true,"F$)1]6{~(SKUfTbwmVWma(rR`jHYTl,=Sh,}a]tN[5jFS9+S$-r)Mef2P4LwK{31_+}]3A].$F$ceo0}^EO:.;,E)C6/sD[W&7,$H(tkwN"]
]
];
# end of cocycles
LOOPS_cc_bases[5] :=
[
# for factors of order 5
[
[1,2]
],
# for factors of order 25
[
[2,1,7,6,9,4,3],
[1,8],
[5,3]
]
];
# end of bases
LOOPS_cc_coordinates[5] :=
[
# for factors of order 5
[
" 1 5 6 7"
],
# for factors of order 25
[
" 1 5 6 1Y 1Z 1a 1c 1b 1s 1t 1u 1x 1y 1z 1$ 1# 1> 1? 1] 8M 8N 8R 8S 8m 8r 8/ 8\
> Z& Z' Z( Z* Z) Z_ Z` Z{ Z~ a0 a1 a3 a2 aJ aK aN aO aP aQ aS aR ai aj ak b8 b\
9 bA bC bB bN bO bP bX bY bm bn fJ fK fL fN fM fT fU fV fk fl fp fq fu fv",
"1 5 6 7",
"5 6"
]
];
# end of coordinates

186
data/cc/cc_cocycles_7.tbl Normal file
View File

@ -0,0 +1,186 @@
#############################################################################
##
#W cc_cocycles_7.tbl G. P. Nagy / P. Vojtechovsky
##
#H @(#)$Id: cc_cocycles_7.tbl, v 3.4.0 2016/11/2 gap Exp $
##
#Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary),
#Y P. Vojtechovsky (University of Denver, USA)
##
# cocycles for CC loops of orders 49, 343
LOOPS_cc_used_factors[7] :=
[
# factors of order 7
[1],
# factors of order 49
[1,2,3]
];
# end of used_factors
LOOPS_cc_cocycles[7] :=
[
# for factors of order 7
[
[false,"3_pn#v%vZG2u,hm"],
[true,"8XD)^4a"]
],
# for factors of order 49
[
[
false,
"6LZ7zqxgc}c-lilPiC`M?GB6K:3oHaifw+m}Wck0)$<:23`8G]^`ZNWi.jX.PD)LQ=t;P@z-vmnQb\
u&[)h]#pmnh/:4PM4aKzardQ)8`}Ki'#/vc1)8#KUEiQ#^|_2tjm<2C&v2kclaZME~QSv5f~AW9LNK\
[7XE_.[<UUMJAWl4(8N7%Ul(F(N'nbOA^0TLUSd,uRYf4+Y~~4LR_{%#7YVaPq37CIb3isd@Ieo6|@\
LIO.C:EvqZ^7QPGo)QI@D0MnNK]}~,LD]K,dbaWPBsO^?'yl{DJs_)Vk}ENo]t~+z.rwX:>/HfA(c'\
Hsao|DL7]'aNjxMcBf?R83)6}/&E|&%(]yXE0$GT$^6dn~(1c:^83vKGaC{:U-9(tbLhd;4:s=;xg'\
lICKDy:=4&Ne5_XUb0Ev:C@e5UySPMu8-)X,_s.Q$<R]P){BhxNgi318K+O&rfWjU}j0&T`gX5B#_p\
Y{q6b]1pX@7pN9(%s&hm}*:kcj(L,we,cr(Dlo8jMO_+4/nA~KXrWb:Mk{X6xgSzqKshvqi5Yiw;S1\
Ca}QhamX6bXo[B06*e0n1w?46:'`<Z?QL|/`#$gK1#:H)>PSt{O+nQ(cSq?l9aN/W2M2)6IWhbv{qU\
Kmh]0tEn/;Ieb}wlAsgjtdm9kn3hIK;78-8QRG^8KSy1S9)k2hfL~0q>{{wmZ~(2HTGvdx62e1MJ)6\
@diIE<u#Q&sDhq:~9H:HJ&?z2Q&<D+F975-#"
],
[
false,
"2<#MOMPOPC;<p:+BY.<Jr#4=PDNo@BsU&P01K>)1$ih(%6g9W1*jYw;CH/BS|06d::-u8A;nYily<\
^hsnNE=ju8x9SHanqD<W#(5+.$)^rX9znN#E$><y??2l01ID.tQ/8wLnjC_C@X^[us:U:y8^)crMuC\
><1KaeN2fRVA>ci(ZND;lDo@bQgpzn=kH)p@8@|Rhd)76f[{/,VefZHGAhn(4|2-fJ}G1Lp}w#[R<*\
TSx;TF~;O~uYE',oTd:m.l;#$h1)Gj=nQI8Qv}mIHax7|JQH0-kj<LW{vB%d,'Z7gntZ~;b7(z9*z9\
5;&>,lXgy6/%;Z3F8>gi<+4Y;RWK=faq6^0b&J|'ngj/e,jT~.$D&A[27.ULiB`WBIp,IX6d@Oex%'\
amVp?r~2r|5~`*;lJm)&@IYmigqxY|~-K-Ju(O,/0hkpSoQ`0r^T#Py`xrY]MKxplgp7':O>5s'C0r\
:^]bRm%]xH7nwqo:NGnSXuVMD|)DgRD+XIdKb5EWWUz>#3)nC(T#-+w@1Q%DP/Ar%oZRVrczCnvrdR\
4?3LogK&@[8tyrH2Dx{;&Uqz$@m(d2W/,8E8FiEQwMLXf9JCNiv(iE/uFwZa((uqZWR082uWX|U9%2\
T4$Zx<tx<%^W;>0RdMR=6uS[qH~Atk(])>;i7pv;lzT*Xnm|/PAO.jF&Z_?)'9Z:QO'81-%cpZ909O\
;]>wlB{#z5T<8wR|[Q?p@BO>~'4fZ[MR#52^AB<6g/=SA-hS2<PHs'WH{c<jEDjJ1G5?D>xo)iQ~hW\
uOnR@*}ei~(^1pew*hG:1R.ph`w(@Ea+11_7mI@pM^9(3Drdu2%e8f_8E3uW-7ao<#0&_Sz;pb>SKJ\
cH-g8T0"
],
[
true,
"1J-4@vr$laUP%E'f}fdk{mJQ<8*/FneW<:hSbou9tgvHHr.B1Dd'[=1#dTe)3b^8uZtpzrH]>f'KT\
k1:pG'dC{}<D;KVDJP#VgM:P2=wWOFloiFiG8#oK:DFI5xV$B7jBS]-RB[2D9<09U;DWjMz%&.l<)[\
>q0}Tdgh[6W5@A^&*yhDvxg=HeC>=#pkYBJ[`Wc8;Ob_sWg[/jv*sGOk2ppB.T5&79LTsTUS4[>9zb\
HgJ0)AQH$Sn{EnI]cCjT.t)}9<0=+(VXm):8*+{o;(59oO6ux.RTdif=~{DVFzOOqmKv-h~`ol{c,C\
vNx%7yF2GMF*stz-A/.vmvz,K?JWkugI|eLJJ/%#C,GU-C_1=WW$[Flo)mzD:Tn8"
],
[
false,
"|5O8_EKy9j'.GTMBs85b%PD&c/?MZ'w?C.~]frDM$vZ,G^F22^.tosRoj3~}A$D+;yYQDA|xZzpoM\
<inE-VPi%[akvH,;5[=XNT(,IzV-l;`$l0(E/<x,Jd'RwQ=O/}ZK+C]OeEkNA|=Ast[O@Jy5W3'di:\
)>2=jgs0g$Yn$$LYJZ$?kALgnqY~)`yo=p'RY9ap58,Q$1no/@ez6YzIjB{f/,qN:I2m*BL[Ng7s4d\
{o=d*)K<-%0D}eX+m=zg4EuAegss_~f={Y`A/{Lp)3Z8;Rl+%h&`,e'+Ig~,b8J_ZU49%f2p[vh:l&\
qY|IhRb=chw+N%_fxA`UkQe2pF#eWUe<rHK]0`@sCtvA_Xi%kKAjOvkpy^,2d~H)gOw#Gz]57>/uzQ\
5l&}}QFv+>T</7uJRHb=LN*spapGLoJI;:-mKv_2+x>,%[c85_CJL`l_Ps?$M0q6YDIwMpaE~F#Tk^\
QKtFlR,pr@#J^F}jtFcMrx20sTWeT-s&BuToQn6W/a`U_tI1cxc+@s/TGf,AHM;SFmM=QNNe`oQThQ\
*+L;qWiGtz{52dZJ{*=]7V/Elmm{nm'8_HG|on|5:tm)zEzBsGe?.;/3s6KqNqKb0?|-$h(G'f-;I0\
xJZsV.b/:FNy.2,$mF@4R`D/H{27rY_fj:sXxhI&^M<N=SWKV.;GP$ag<XJVN.[0(3.u)`|tX^~)7[\
><@-P_,~l7A(h?9I[O*Ql9<Y>Ap*DXoqp-lzr<HiA'dAZiDHk{Kua>b{g#UK_L>OdW:s8[8d^Md8R>\
ZE;B^gq/%jO0hhvH6B:`EXYPx,Y64+>IFCOF$C}uD(c]ACzID<zc0|y^-t3KIHrQ2(hcZ~B&%w`ycE\
872Js}_:#r'/?RSjeU>(b<Ho^Co.q_"
],
[
false,
"|5O8_EKy9j'.GTMBs85b%PD&c/?MZ'w?C.~]frDM$vZ,G^F22^.tosRoj3~}A$D+;yYQDA|xZzpoM\
<inE-VPi%[akvH,;5[=XNT(,IzV-l;`$l0(E/<x,Jd'RwQ=O/}ZK+C]OeEkNA|=Ast[O@Jy5W3'dod\
<x:&uNKwLO<,6vBY:PIaEN.#Y#LLnW'-W5GO#rp(Kt<,V[4hC^Cv4_iy<wNP4]w3q`P,/YY50{c+AJ\
|%AssRR+}[NR}3kx%5(ybiJxt:.`k]7|(K3P,c?[FQa&}]]Btj0%S+MP*Obv;I}bp6VtGk0X<6JCU4\
xgQ$L>*Oz1JSPfnyQxyTIVj(grQ(.{]<_QNd{HvB>cvo)pQ0'`_K2LOXFP;zqY/Ucl_vuqC2dsvw*g\
-4EaN1U28ip],fviGs2tDw7b60Wn-=b`1++TH7guS$O&V|uP`AO5H|3MxWG>r%pH9OX|4uvRJNG;ql\
?jRsTg`>.rC}'4G67|BGe:YFYyqe9/BAZk^En]#4Z;n|(7q/<bXGlM+o.#^b:1,/]y,3ajw3V)dMc/\
{K5LvfFziHK(;s7Cn|wmaP$+2FUfS3dzB~PHaQ#ZZ2Ogk}OZ+:R|`YyesFa~K3S'0E`Gwv3~A&>eyS\
G[JZ(Xi9H~Y(<U:.EU#)jfW%*{[<jJ&{8]^<.k:x]ujifa6H2X?Ryx})Xa*7li;lhAM(q_0g^m<G<@\
8sb,a9^kBCo,g6:C||I|XV`2V('7g{MZikAo9Nd`Ga+G1Akiq?+i~a$E>q]$KS,3niWslVp`h2O@@^\
5k4UO4C^SVuX8JfE(un?-2`3|o`U$'i0Z`Tl^T0&w=RAS?,DPC^]Lm>#I&mEH2hq.2qRnClYht]=Gi\
i,_4/QLlTY*hs+;2Wx6-QIA}'AHv=k"
],
[
false,
"1R<zJxqLZ48F40Z6^DE;>A2VlizlKV3v-M)ZvkQ:Tji6)q?.aHY/DVViT^./4LJWh`rt`xi[*&<Rl\
?]E6^]ZDb&la7XyL+xNM97c2?SV8=y`UBdVNRs$0|6ibjenX;[g-3Vu7|#DMb+sm^6XN8_>9|aTH?L\
du#P07`Iv7CTiny~&/LQB>+#Zl9=NhGgUx]]#w;-GQ'wv]/FjC|4^7ZT%{zR>2rI(V/y.h([}$t2iz\
f-;OMY&e3Psmid9?So^iLiPu|gJ&VjM]s-=+Eg|[eE]7+T3[P>Svq+jFh>Kh5xs<_7Ck;p|]_#'iYy\
'X[Qy>.gZduu'u+2T)nm3u?%(29||or}k`X|v%h%J#/J4,3L&X00^aQ{;?;|'Pb.4Eht7Ca;5.S[j`\
`0c`V]Ub;<ae=}#FCIO@EZ=+G1|#~SD:<'^P2;UQCQ_ub&XpILaQCR[+oSevh/W+}-Er92E#:?t#@f\
VBn]*{k;jTT{8RT5R;y+z]bJDP_9nn]d8fL|h^)zwCpj9_7-rl2|U1j@~>:5Q`b[[M-Oh;(@tF[+Tv\
eHb>qs}nAl3y7%HqX1=?%-n0i*ocR*xBZ_x$;4x]0Fu7^lEB~k>c?[(`QDRjVX]_<2ij}IFx#<9K2-\
@)dqWB~HzSZu]FO]({=(2[~LxX`Lt;t}I:h=m$qC?GcH2$TbtMKF9X3}69Z((tU]u|Tncz:v|F|UnY\
egR=5{).>w)nn09;eYNX;P`bQ/v%#YU=wtMH^/BX{'X{AuPd]kn;J[o;ekyo[_:E0r]mYYeWV>s,7B\
BE]?}>*N8fb&bz%TB}}.q#zQmGe+N%6>ZW'9C&j?h0}H/MA3,Q%fs<69Ugp<$35yuSsYjNi>uW4+*F\
joC{o}A*aMgY6B~K-%)s,M0cMl+RyG?Me&*0<tn@Y;$,06Z.';'I$dS>NXHhNI[wPKmAAxi=$[eKf'\
5wm;j4Ia&t?o}.7PC[8p0`}m1K7Z=6i0AaV3|Cx8QHUtN@s_d_5/Wxy?"
],
[
false,
"Lw{bh?7j,xG$S*E6FG+~*YFNS.3R1PZFV*O}$x=$,>L<J.Yav<n_Wd5ndc~;El}*M/xbtV'08T/fa\
$'$-#X{7dM/H*$.V&S$B>#)im:wsCs<Y1sSE*&SUkV#YJqmB$8J0mqHelt3+1qN${_laewI`^uvFA9\
(I5<90(I/h4Ws0(3d?J&+[UJ'~jBPaAM4&vT}b2#18r%{Qm'_Ah+t4A(6qI^#%OV7s9_Vil?8ViR+z\
f[TF->eS)/+@9A)t[n.3|dKa.@K`s)}Nsu*^;9StpOA[0(XB.V`P.~L0v}d1+^p2NB)g26p^Psb{V9\
BHO_{bg/p@`pL7IZ$b6>,yb-t;B'08oMD&Y$JO'kC$rXI~$AhF8[@PHwH3c$14p-O7l7B+[ROu$qWy\
{^Q6C>Kfp/GF;9XW6KK=oy[}4~YcJ.|)%9UuHh'pWFfpeqQ]GJOOM`-8ED_B7bQCDakbBNmL'OB;5j\
;{T}<fQ;3+(nR3TOW:PWIvy#StF:orsq/TjSwj)?Y:5Ca6d/hp,(w#IAJ7zH^`)pIA@jwO<hZ??s>A\
Ld:Q0HyXS}I/8L3uO.J$jEe'h~K_<OY]G{4HtG(,[v=?@#84%MIm$Sqs?nUi>=4&-5w=z|;emc@CrC\
jUR>)us<r{'6uz=Ab'A;,}DazUn,ZdwOQIQlF&ni>E~*56m#tvuaVadVoViN4e&+pJyoS%2Fm;Q[2x\
K)3-MvYx(5tu>t6l~%HaS)y]T@/:.pHpozrwa_a0qU2pj/S%r?U1-EL9/if(r^h=b-|F1f-PrEPsz_\
}GB52uB'I`u84~gO<[8/vkfDCki`;5.A[p9$d?xM(p3GWRNJ#@Wq)yLzBZ(q)`bEEHrw8ZKH*'Vw0h\
[lL@121{+P?zf}OE$YsQkfuH[tIZ?.{H$1D7|C&W7BMTqfSy<gl&C^1):1qdXEhCi5bygN?TgD~$`W\
-*dY'6rOu3i],t2P*;L*AoJJpG('I7#]cZ'%arVh]]tl`{4ooc;P00@l7N"
],
[
true,
"PqiHcf~[]i1P_1|@t$o>+d9Ecilr7kNxGB;R[w/aw.3O[%>aeU=WJ~Y8O$zGGV^coY//F1eor,:Td\
-Km_[4gn:qN%,Jjh%=j5*|60k,I=R0AoUy/AjApFR:YiCN(hs|<(9oz38k:*P8vb}.s-V%jKa?AtGG\
5+[_~.e@9vp_G1+AS'SjrvnBt22Oxe0~*ce|@N./@PJK{w%JRq=Ur~R`fn)PB3MS|Yk%ObQ}(nt9VB\
}_H.s$9'U~:olU@Nr[}PIn@{@WP~#8gW{sY+YD*+J*en&,U;L>rks(<{7FkONs4sLsuAqXL,tS3C8I\
|3DaDkb<A3V6F)D+e~3B]0X~^h)PF|$BF]WVZvxECa0c?6k6yT:g*>dM6vPb1:`/S}%V?_yqzxZ$+v\
dltf>v2Y1DG=ud+yJHu&UcNXnWr)x9aO2_'rDV(CtY2XwuYYuZNTVMSJIk{(/-e|R>AK=-DJW]YkWr\
SRZx#xA)R&~9,|@H$;Qp44%1OL=[)nVY-T#e?="
],
[
true,
"PqiHcf~[]i1P_1|@t$o>+d9Ecilr7kNxGB;R[w/aw.3O[%>aeU=WJ~Y8O$zGGV^coY//F1eor,:Td\
-Km_[4gn:qN%,Jjh%=j5*|60k,I=R0AoUy/AjApFR:YiCN(hs|<(9v|,VJWuIaSjnjND:b{ySC9.H|\
r$]Z..3Z0:[?2PaK`}A]B>uB*h~/>1M0uD<)De=|9d+pOn?]wErg1[*0-R5)?qblH%`73nvNmZQ;z5\
.WX?J+odUr>4fY`~cqv}.`(%zI,UZ@/P1{tAm<q`~9Kqo/|?wZl<_Z*uTo|o&O{gb_bV$|l5PeO7Bg\
%Ffgx_$>Y}-GwoXn)$pvic%DKA:<:bJ.IQPSnnLwuWDD,]5{([)XEf1M,5<vM.Ru)f7n%8HDv_xM]V\
8f@7iO;>ORc<D6P.;pLjwMp0JJhyq:Jb./,/3iQ`^8^QYrBG#MQj}ZjmUu3nmZ-4gwWoqztIA>Lha*\
aD[J{@-Zw)BE`cte3}[*+7)J:0YITvzb58xr*g"
]
]
];
# end of cocycles;
LOOPS_cc_bases[7] :=
[
# for factors of order 7
[
[1,2]
],
# for factors of order 49
[
[2,1,7,6,9,4,3],
[1,8],
[5,3]
]
];
# end of bases
LOOPS_cc_coordinates[7] :=
[
# for factors of order 7
[
" 1 7 8 A"
],
# for factors of order 49
[
" 1 7 8 3+ 3, 3. 3- 3; 3/ 3: 4E 4F 4H 4S 4T 4V 4U 4Y 4W 4X 4$ 4% 4' UE UF UL U\
M U% U, V. V? 26g 26h 26j 26i 26m 26k 26l 26< 26= 26? 270 271 273 272 27A 27Z \
27a 27b 2E0 2E1 2E3 2E2 2E6 2E4 2E5 2E7 2E8 2EA 2En 2Eo 2Eq 2Ep 2Et 2Er 2Es 2E\
u 2Ev 2Ex 2Ew 2E# 2Ey 2Ez 2E+ 2E, 2E. 2E- 2E; 2E/ 2E: 2E< 2E= 2E? 2E> 2E] 2E@ \
2E[ 2T7 2T8 2TA 2T9 2TD 2TB 2TC 2TS 2TT 2TV 2Tx 2Tw 2Ty 2Tz 2T& 2T( 2T? 2T> 2T\
@ 2T[ 2T- 2T/ 2U% 2U& 2U_ 2U`",
"1 7 8 A",
"7 8 A 9"
]
];
# end of coordinates

94
data/rightbruck.tbl Normal file
View File

@ -0,0 +1,94 @@
#############################################################################
##
#W rightbruck.tbl Right Bruck loops G. P. Nagy / P. Vojtechovsky
##
#H @(#)$Id: rightbruck.tbl, v 3.4.0 2017/10/24 gap Exp $
##
#Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary),
#Y P. Vojtechovsky (University of Denver, USA)
##
# Right Bruck loops of order 243 are represtented as central
# extensions of the cyclic group of order 3.
# The necessary data consists of:
# - LOOPS_right_bruck_cocycles, a basis for a superspace of cocycles
# for all factors
# - LOOPS_right_bruck_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_right_bruck_data := [
#implemented orders
[3,9,27,81],
#number of nonassociative loops of given order
[1,2,7,72],
#the loops
[
#order 3 (Z_3)
[
"201"
],
#order 9 (two abelian groups)
[
"204537861534867678012861207201345534",
"204537861534867678120862017012453345"
]
,
#order 27 (placeholder)
[
]
,
#order 81 (placeholder)
[
]
]
];
LOOPS_right_bruck_cocycles :=
[
# cocycles for factors of order 9
[
[ 9, true, "2'pwq" ],
[ 9, false, "ZBj[+>~vni@o7" ],
[ 9, true, "3#QhSwvJx" ],
[ 9, true, "JP_g" ]
],
# cocycles for factors of order 27
[
[ 27, true, "3@1Aju?P%w:;@a,0x-#3WF%=<0|[/XeKIur8%Myq" ],
[ 27, false, "~r]%_p(/($qr{(,=pRww5eIAxf6AV2ya(q-<Q`=n*o?{:aQFYs^Z67D/j;fk~4cBp:8U7M)C|fm@FqBGk,iDTH89oS54Uw9{:O0CIT;,eT`V*E8$nTS/($" ],
[ 27, true, "12.W./ADP*t:.tkW}cA50,w,lViI;q+2X/xlnaF,#}?)usJ9w~~kTglq[qJ~&]L+JcPn<)2OxD1F82f{73:MP(" ],
[ 27, false, "vcV(GHJH0z&v@Sob,UDrLY,WD58fYB|3t0OVb*SbDq,Os1y;[wh2$(r^(U^m@Yz*0EJwD_MM7p/:+?DN|B6@}lcIo#9_Tq,qrV=`$yv;#Ao$@E&ez3zv4WCL/0U%Z=r1*MmY9WL*_rMtFFIWg-9x>" ],
[ 27, false, "AZDm2AKcJkMsQcdWFt+aLX[xZ(r8UY1'fG3@c4[6W~[>TmU0@q1WmzP%~/s[81]q,Q4%z9D/f{Aw;|D'4I3JJA8]#SJw/2{zF*.$LfX)p:H`d4PJpjh5.Ff13Ri*~8y-)#nE&R(U#=p1|MkWN'StZ4n]d?x<_h>7aa,k" ],
[ 27, false, "1M,aaVut'THt&#I8$eX`/zd6Y61SWR*J/0w}.GTnhbAq*[xNs%%~,8d06kof'{(&R;Yp7V^N7}-CM^-xbO@'FOJN$,xA5[(Y-t~7#u>6`}5xnm`[`w&zEv2CDzM/)'l~qj|B*5n[~*p/LX`2bhF^Ni]^|%r{XWec*B)" ],
[ 27, false, "92DM|aq*TP]rsR*EQ$z6je~2'=+_OdSB-n=+e+sX|h9%<-GHVfygl4=MRL`~Z1%WDlExwz:Pkf|&l#$,;gkB~%&6>IN8g9l,W>w0>i^lHKrckmJKMW}vB&" ],
[ 27, true, "2Fn}rAZjS5%1++ZJQ=]$[rHW:=|k`14E44FV{vleOG?HDkycql}DTzo{rNK{449-dI|`]P$~d" ],
[ 27, false, "3No|.dC2S{f$z41B,@,dWo#Pbn^j05+;^BdD%qT#J-Y0A3aX;~]~?]AJFrZ]Lc6,F%(0lQP;2JWtRkAxgwm5hqDjEaOxuA@)*UEuS1i_|neKW>]Gc^f$*hriOT&$W{H5@(Q53QHK$/+BBuv@7Nt57S:GX$#^" ], [ 27, true, "RL7.:Q{l+0K.vFgjWJ>:Al+5-'HV|x-3W6oKPe}" ],
[ 27, false, "S||kz0&UY#`u{sajCjc?q~9uh`&f8x~qY{I{&?E%/NEhWR%Rrs~'rJp7IVUF)rN7Kpk'hTt5)4nz#*IlRw>tDb1TPF[C.&FdB)<(;#XCDP.M/[rnMcXC{j>6:+d6rSyuLodfc*Y=" ],
[ 27, false, "G`EoVUtm{j0}G'NaLnI`~6j6Ihy#JaaR,E`T%O?v--xeZ[QR/sU4VS*ln./35g[m+|KE(R8>HFEuK%~/{<zXhJ93?q46Aa{|;$sjmJUof>//(PsGRw&tu;d4ap8m=Bm<8r&wu^4gQq'V_NXRbh6p#=$1uH$0" ],
[ 27, false, "(m6f.dLaPEH6f9>wYlYw'C=3nMc4C~0l1VkV8t#tpU#m}aQ=/PZv.Npk9aeW4.edR=ISNyLR]dKN?kb#EYl,z0gM^m4$#Fd0nQI5I+P|LWT5wSmhg`9o0" ]
]
];
# The first character of every coordinate determines the factor,
# the remaining characters coorespond to coordinates with respect to the
# above bases.
LOOPS_right_bruck_coordinates :=
[
# coordinates for factors of order 9
[ "1", "1R", "1a", "1C", "1d", "1m", "24" ]
,
# coordinates for factors of order 27
[
"1", "1%G1", "1_p#", "1Slp", "111$q", "11N8M", "1N*|", "1{_}", "11)M?",
"1U@=", "1|(7", "11z]8", "11JB)", "11]R*", "1j`(", "11elc", "13Fa",
"1(Vb", "11QNw", "11lxS", "1ARQ", "11XZm", "1mU3", "11Lk4", "1tf^",
"1#r/", "1`W,", "11xm-", "1MG+", "11?^g", "1hqe", "11G)f", "11jP1",
"11'y#", "11N?V", "1o%U", "11.-q", "11cbE", "11NeQ", "13Nb", "11l(T",
"11=wG", "1fQE", "128|k", "1@&>", "14*i", "1)_j", "27CH", "281",
"27KI", "22z", "2A#", "2I$", "2GB", "27SS", "2Eej", "2I-", "31bWB",
"39", "3%GA", "31bY-", "32+", "41&>u", "4Stt", "411,u", "423iK",
"41bmD", "4S$&", "411>'", "41'4(", "6%GA", "77B`"
]
];

61
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@ -6,7 +6,7 @@
Computing with quasigroups and loops in GAP
Version 3.3.0
Version 3.4.0
Gábor P. Nagy
@ -28,7 +28,7 @@
-------------------------------------------------------
Copyright
© 2016 Gábor P. Nagy and Petr Vojtěchovský.
© 2017 Gábor P. Nagy and Petr Vojtěchovský.
-------------------------------------------------------
@ -167,10 +167,12 @@
6.11-3 QuasigroupsUpToIsomorphism
6.11-4 LoopsUpToIsomorphism
6.11-5 AutomorphismGroup
6.11-6 IsomorphicCopyByPerm
6.11-7 IsomorphicCopyByNormalSubloop
6.11-8 Discriminator
6.11-9 AreEqualDiscriminators
6.11-6 QuasigroupIsomorph
6.11-7 LoopIsomorph
6.11-8 IsomorphicCopyByPerm
6.11-9 IsomorphicCopyByNormalSubloop
6.11-10 Discriminator
6.11-11 AreEqualDiscriminators
6.12 Isotopisms
6.12-1 IsotopismLoops
6.12-2 LoopsUpToIsotopism
@ -256,28 +258,31 @@
9.2 Left Bol Loops and Right Bol Loops
9.2-1 LeftBolLoop
9.2-2 RightBolLoop
9.3 Moufang Loops
9.3-1 MoufangLoop
9.4 Code Loops
9.4-1 CodeLoop
9.5 Steiner Loops
9.5-1 SteinerLoop
9.6 Conjugacy Closed Loops
9.6-1 RCCLoop and RightConjugacyClosedLoop
9.6-2 LCCLoop and LeftConjugacyClosedLoop
9.6-3 CCLoop and ConjugacyClosedLoop
9.7 Small Loops
9.7-1 SmallLoop
9.8 Paige Loops
9.8-1 PaigeLoop
9.9 Nilpotent Loops
9.9-1 NilpotentLoop
9.10 Automorphic Loops
9.10-1 AutomorphicLoop
9.11 Interesting Loops
9.11-1 InterestingLoop
9.12 Libraries of Loops Up To Isotopism
9.12-1 ItpSmallLoop
9.3 Left Bruck Loops and Right Bruck Loops
9.3-1 LeftBruckLoop
9.3-2 RightBruckLoop
9.4 Moufang Loops
9.4-1 MoufangLoop
9.5 Code Loops
9.5-1 CodeLoop
9.6 Steiner Loops
9.6-1 SteinerLoop
9.7 Conjugacy Closed Loops
9.7-1 RCCLoop and RightConjugacyClosedLoop
9.7-2 LCCLoop and LeftConjugacyClosedLoop
9.7-3 CCLoop and ConjugacyClosedLoop
9.8 Small Loops
9.8-1 SmallLoop
9.9 Paige Loops
9.9-1 PaigeLoop
9.10 Nilpotent Loops
9.10-1 NilpotentLoop
9.11 Automorphic Loops
9.11-1 AutomorphicLoop
9.12 Interesting Loops
9.12-1 InterestingLoop
9.13 Libraries of Loops Up To Isotopism
9.13-1 ItpSmallLoop
A Files
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>
<p>Version 3.3.0</p>
<p>Version 3.4.0</p>
</div>
<p><b>Gábor P. Nagy
@ -50,7 +50,7 @@
<p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p>
<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>
@ -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#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#X85B3E22679FD8D81">6.11-6 IsomorphicCopyByPerm</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#X7D09D8957E4A0973">6.11-8 Discriminator</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#X7A42812B7B027DD4">6.11-6 QuasigroupIsomorph</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#X85B3E22679FD8D81">6.11-8 IsomorphicCopyByPerm</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 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>
@ -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#X8774304282654C58">9.2-2 RightBolLoop</a></span>
</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>
<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 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>
<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 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>
<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 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>
<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 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 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>
</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>
<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 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>
<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 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>
<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 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>
<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 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>
<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 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>
<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 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
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
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
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.
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.
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.
Katharina Artic. The library of right Bruck loops of order 27, 81 was
obtained jointly with Izabella Stuhl.
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

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@ -73,7 +73,7 @@
<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>
@ -127,9 +127,9 @@ gap&gt; WriteGapIniFile();;
<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>

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

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@ -72,19 +72,19 @@
<pre class="normal">
DeclareCategory( "IsQuasigroupElement", IsMultiplicativeElement );
DeclareRepresentation( "IsQuasigroupElmRep",
IsPositionalObjectRep and IsMultiplicativeElement, [1] );
DeclareCategory( "IsLoopElement",
IsQuasigroupElement and IsMultiplicativeElementWithInverse );
DeclareRepresentation( "IsLoopElmRep",
IsPositionalObjectRep and IsMultiplicativeElementWithInverse, [1] );
## latin (auxiliary category for GAP to tell apart IsMagma and IsQuasigroup)
DeclareCategory( "IsLatin", IsObject );
DeclareCategory( "IsQuasigroup", IsMagma and IsLatin );
DeclareCategory( "IsLoop", IsQuasigroup and
IsMultiplicativeElementWithInverseCollection);
DeclareCategory( "IsQuasigroupElement", IsMultiplicativeElement );
DeclareRepresentation( "IsQuasigroupElmRep",
IsPositionalObjectRep and IsMultiplicativeElement, [1] );
DeclareCategory( "IsLoopElement",
IsQuasigroupElement and IsMultiplicativeElementWithInverse );
DeclareRepresentation( "IsLoopElmRep",
IsPositionalObjectRep and IsMultiplicativeElementWithInverse, [1] );
## latin (auxiliary category for GAP to tell apart IsMagma and IsQuasigroup)
DeclareCategory( "IsLatinMagma", IsObject );
DeclareCategory( "IsQuasigroup", IsMagma and IsLatinMagma );
DeclareCategory( "IsLoop", IsQuasigroup and
IsMultiplicativeElementWithInverseCollection);
</pre>
@ -162,15 +162,15 @@ DeclareCategory( "IsLoop", IsQuasigroup and
<p>In the following example, <code class="code">L</code> is a loop with two elements.</p>
<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">L; </span>
&lt;loop of order 2&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Print( L ); </span>
&lt;loop with multiplication table [ [ 1, 2 ], [ 2, 1 ] ]&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Elements( L ); </span>
[ l1, l2 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">SetLoopElmName( L, "loop_element" );; Elements( L ); </span>
[ loop_element1, loop_element2 ]
<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">L;</span>
&lt;loop of order 2&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Print( L );</span>
&lt;loop with multiplication table [ [ 1, 2 ], [ 2, 1 ] ]&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Elements( L );</span>
[ l1, l2 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">SetLoopElmName( L, "loop_element" );; Elements( L );</span>
[ loop_element1, loop_element2 ]
</pre></div>

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@ -188,15 +188,15 @@
<p>Since <code class="code">CanonicalCayleyTable</code> is called within the above operation, the resulting quasigroup will have Cayley table with distinct entries <span class="SimpleMath">\(1\)</span>, <span class="SimpleMath">\(\dots\)</span>, <span class="SimpleMath">\(n\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">ct := CanonicalCayleyTable( [[5,3],[3,5]] ); </span>
[ [ 2, 1 ], [ 1, 2 ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">NormalizedQuasigroupTable( ct ); </span>
[ [ 1, 2 ], [ 2, 1 ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LoopByCayleyTable( last ); </span>
&lt;loop of order 2&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ IsQuasigroupTable( ct ), IsLoopTable( ct ) ]; </span>
[ true, false ]
<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">ct := CanonicalCayleyTable( [[5,3],[3,5]] );</span>
[ [ 2, 1 ], [ 1, 2 ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">NormalizedQuasigroupTable( ct );</span>
[ [ 1, 2 ], [ 2, 1 ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LoopByCayleyTable( last );</span>
&lt;loop of order 2&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ IsQuasigroupTable( ct ), IsLoopTable( ct ) ];</span>
[ true, false ]
</pre></div>
<p><a id="X849944F17E2B37F8" name="X849944F17E2B37F8"></a></p>
@ -235,56 +235,56 @@
<p><strong class="button">Example:</strong> Data does not have to be arranged into an array of any kind.</p>
<p class="center">\[
\begin{array}{cccc}
0&amp;1&amp;2&amp;1\\
2&amp;0&amp;2&amp; \\
0&amp;1&amp; &amp;
\end{array}\quad + \quad "" \quad \Longrightarrow\quad
\begin{array}{ccc}
1&amp;2&amp;3\\
2&amp;3&amp;1\\
3&amp;1&amp;2
\end{array}
<p class="center">\[
\begin{array}{cccc}
0&amp;1&amp;2&amp;1\\
2&amp;0&amp;2&amp; \\
0&amp;1&amp; &amp;
\end{array}\quad + \quad "" \quad \Longrightarrow\quad
\begin{array}{ccc}
1&amp;2&amp;3\\
2&amp;3&amp;1\\
3&amp;1&amp;2
\end{array}
\]</p>
<p><strong class="button">Example:</strong> Chunks can be any strings.</p>
<p class="center">\[
\begin{array}{cc}
{\rm red}&amp;{\rm green}\\
{\rm green}&amp;{\rm red}\\
\end{array}\quad + \quad "" \quad \Longrightarrow\quad
\begin{array}{cc}
1&amp; 2\\
2&amp; 1
\end{array}
<p class="center">\[
\begin{array}{cc}
{\rm red}&amp;{\rm green}\\
{\rm green}&amp;{\rm red}\\
\end{array}\quad + \quad "" \quad \Longrightarrow\quad
\begin{array}{cc}
1&amp; 2\\
2&amp; 1
\end{array}
\]</p>
<p><strong class="button">Example:</strong> A typical table produced by <strong class="pkg">GAP</strong> is easily parsed by deleting brackets and commas.</p>
<p class="center">\[
[ [0, 1], [1, 0] ] \quad + \quad "[,]" \quad \Longrightarrow\quad
\begin{array}{cc}
1&amp; 2\\
2&amp; 1
\end{array}
<p class="center">\[
[ [0, 1], [1, 0] ] \quad + \quad "[,]" \quad \Longrightarrow\quad
\begin{array}{cc}
1&amp; 2\\
2&amp; 1
\end{array}
\]</p>
<p><strong class="button">Example:</strong> A typical TeX table with rows separated by lines is also easily converted. Note that we have to use <span class="SimpleMath">\(\backslash\backslash\)</span> to ensure that every occurrence of <span class="SimpleMath">\(\backslash\)</span> is deleted, since <span class="SimpleMath">\(\backslash\backslash\)</span> represents the character <span class="SimpleMath">\(\backslash\)</span> in <strong class="pkg">GAP</strong></p>
<p class="center">\[
\begin{array}{lll}
x\&amp;&amp; y\&amp;&amp;\ z\backslash\backslash\cr
y\&amp;&amp; z\&amp;&amp;\ x\backslash\backslash\cr
z\&amp;&amp; x\&amp;&amp;\ y
\end{array}
\quad + \quad "\backslash\backslash\&amp;" \quad \Longrightarrow\quad
\begin{array}{ccc}
1&amp;2&amp;3\cr
2&amp;3&amp;1\cr
3&amp;1&amp;2
\end{array}
<p class="center">\[
\begin{array}{lll}
x\&amp;&amp; y\&amp;&amp;\ z\backslash\backslash\cr
y\&amp;&amp; z\&amp;&amp;\ x\backslash\backslash\cr
z\&amp;&amp; x\&amp;&amp;\ y
\end{array}
\quad + \quad "\backslash\backslash\&amp;" \quad \Longrightarrow\quad
\begin{array}{ccc}
1&amp;2&amp;3\cr
2&amp;3&amp;1\cr
3&amp;1&amp;2
\end{array}
\]</p>
<p><a id="X81A1DB918057933E" name="X81A1DB918057933E"></a></p>
@ -329,14 +329,14 @@
<p>These are the dual operations to <code class="code">QuasigroupByLeftSection</code> and <code class="code">LoopByLeftSection</code>.</p>
<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">ls := LeftSection( S ); </span>
[ (), (1,3,5), (1,5,3) ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">CayleyTableByPerms( ls ); </span>
[ [ 1, 3, 5 ], [ 3, 5, 1 ], [ 5, 1, 3 ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">CayleyTable( LoopByLeftSection( ls ) ); </span>
[ [ 1, 2, 3 ], [ 2, 3, 1 ], [ 3, 1, 2 ] ]
<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">ls := LeftSection( S );</span>
[ (), (1,3,5), (1,5,3) ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">CayleyTableByPerms( ls );</span>
[ [ 1, 3, 5 ], [ 3, 5, 1 ], [ 5, 1, 3 ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">CayleyTable( LoopByLeftSection( ls ) );</span>
[ [ 1, 2, 3 ], [ 2, 3, 1 ], [ 3, 1, 2 ] ]
</pre></div>
<p><a id="X85ABE99E84E5B0E8" name="X85ABE99E84E5B0E8"></a></p>
@ -360,11 +360,11 @@
<p>Here is a simple example in which <span class="SimpleMath">\(T\)</span> is actually the right section of the resulting loop.</p>
<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">G := Group( T );; H := Stabilizer( G, 1 );; </span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LoopByRightFolder( G, H, T ); </span>
&lt;loop of order 5&gt;
<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">G := Group( T );; H := Stabilizer( G, 1 );;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LoopByRightFolder( G, H, T );</span>
&lt;loop of order 5&gt;
</pre></div>
<p><a id="X8759431780AC81A9" name="X8759431780AC81A9"></a></p>
@ -394,16 +394,16 @@
<p>Returns: The extension of an abelian group <var class="Arg">K</var> by a loop <var class="Arg">F</var>, using action <var class="Arg">f</var> and cocycle <var class="Arg">t</var>. The arguments must be formatted as the output of <code class="code">NuclearExtension</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">F := IntoLoop( Group( (1,2) ) ); </span>
&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">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">LoopByExtension( K, F, phi, theta ); </span>
&lt;loop of order 8&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">IsAssociative( last ); </span>
false
<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">F := IntoLoop( Group( (1,2) ) );</span>
&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">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">LoopByExtension( K, F, phi, theta );</span>
&lt;loop of order 8&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">IsAssociative( last );</span>
false
</pre></div>
<p><a id="X7AE29A1A7AA5C25A" name="X7AE29A1A7AA5C25A"></a></p>

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@ -167,17 +167,17 @@
<p>Returns: The left inverse, right inverse and inverse, respectively, of the quasigroup element <var class="Arg">x</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">CayleyTable( Q ); </span>
[ [ 1, 2, 3, 4, 5 ],
[ 2, 1, 4, 5, 3 ],
[ 3, 4, 5, 1, 2 ],
[ 4, 5, 2, 3, 1 ],
[ 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">[ 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>
[ l5, l4, fail ]
<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">CayleyTable( Q );</span>
[ [ 1, 2, 3, 4, 5 ],
[ 2, 1, 4, 5, 3 ],
[ 3, 4, 5, 1, 2 ],
[ 4, 5, 2, 3, 1 ],
[ 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">[ 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>
[ l5, l4, fail ]
</pre></div>
<p><a id="X7E0849977869E53D" name="X7E0849977869E53D"></a></p>

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@ -488,17 +488,28 @@
the elements of the underlying quasigroup without changing the isomorphism
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
Returns: When Q is a quasigroup and f is a permutation of 1,dots,|Q|,
returns a quasigroup defined on the same set as Q with
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.
Returns: LoopIsomorphism(Q,f) if Q is a loop, and
QuasigroupIsomorphism(Q,f) if Q is a quasigroup.
6.11-7 IsomorphicCopyByNormalSubloop
6.11-9 IsomorphicCopyByNormalSubloop
IsomorphicCopyByNormalSubloop( Q, S )  operation
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
under isomorphisms. The following two operations are used in the search.
6.11-8 Discriminator
6.11-10 Discriminator
Discriminator( Q )  operation
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
have identical discriminators, they may or may not be isomorphic.
6.11-9 AreEqualDiscriminators
6.11-11 AreEqualDiscriminators
AreEqualDiscriminators( D1, D2 )  operation
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#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#X85B3E22679FD8D81">6.11-6 IsomorphicCopyByPerm</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#X7D09D8957E4A0973">6.11-8 Discriminator</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#X7A42812B7B027DD4">6.11-6 QuasigroupIsomorph</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#X85B3E22679FD8D81">6.11-8 IsomorphicCopyByPerm</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 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>
@ -265,28 +267,28 @@
<p>Note how the Cayley table of a subquasigroup is created only upon explicit demand. Also note that changing the names of elements of a subquasigroup (subloop) automatically changes the names of the elements of the parent subquasigroup (subloop). This is because the elements are shared.</p>
<div class="example"><pre>
<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;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ Parent( S ) = M, Elements( S ), PosInParent( S ) ]; </span>
[ true, [ l1, l3, l5], [ 1, 3, 5 ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">HasCayleyTable( S ); </span>
false
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">SetLoopElmName( S, "s" );; Elements( S ); Elements( M ); </span>
[ s1, s3, s5 ]
[ s1, s2, s3, s4, s5, s6, s7, s8, s9, s10, s11, s12 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">CayleyTable( S ); </span>
[ [ 1, 3, 5 ], [ 3, 5, 1 ], [ 5, 1, 3 ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LeftSection( S ); </span>
[ (), (1,3,5), (1,5,3) ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ HasCayleyTable( S ), Parent( S ) = M ]; </span>
[ true, true ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">L := LoopByCayleyTable( CayleyTable( S ) );; Elements( L ); </span>
[ l1, l2, l3 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ Parent( L ) = L, IsSubloop( M, S ), IsSubloop( M, L ) ]; </span>
[ true, true, false ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LeftSection( L ); </span>
[ (), (1,2,3), (1,3,2) ]
<div class="example"><pre>
<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;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ Parent( S ) = M, Elements( S ), PosInParent( S ) ];</span>
[ true, [ l1, l3, l5], [ 1, 3, 5 ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">HasCayleyTable( S );</span>
false
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">SetLoopElmName( S, "s" );; Elements( S ); Elements( M );</span>
[ s1, s3, s5 ]
[ s1, s2, s3, s4, s5, s6, s7, s8, s9, s10, s11, s12 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">CayleyTable( S );</span>
[ [ 1, 3, 5 ], [ 3, 5, 1 ], [ 5, 1, 3 ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LeftSection( S );</span>
[ (), (1,3,5), (1,5,3) ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ HasCayleyTable( S ), Parent( S ) = M ];</span>
[ true, true ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">L := LoopByCayleyTable( CayleyTable( S ) );; Elements( L );</span>
[ l1, l2, l3 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ Parent( L ) = L, IsSubloop( M, S ), IsSubloop( M, L ) ];</span>
[ true, true, false ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LeftSection( L );</span>
[ (), (1,2,3), (1,3,2) ]
</pre></div>
<p><a id="X78ED50F578A88046" name="X78ED50F578A88046"></a></p>
@ -349,16 +351,16 @@ false
<p>Here is an example for multiplication groups and inner mapping groups:</p>
<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">M := MoufangLoop(12,1); </span>
&lt;Moufang loop 12/1&gt;
<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)
<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;
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>
216
<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">M := MoufangLoop(12,1);</span>
&lt;Moufang loop 12/1&gt;
<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)
<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;
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>
216
</pre></div>
<p><a id="X7B45C2AF7C2E28AB" name="X7B45C2AF7C2E28AB"></a></p>
@ -464,16 +466,16 @@ Group([ (4,6)(7,11), (7,11)(8,10), (2,6,4)(7,9,11), (3,5)(9,11), (8,12,10) ])
<p>Returns: When <var class="Arg">S</var> is a normal subloop of a loop <var class="Arg">Q</var>, returns the natural projection from <var class="Arg">Q</var> onto <var class="Arg">Q</var><span class="SimpleMath">\(/\)</span><var class="Arg">S</var>.</p>
<div class="example"><pre>
<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;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">IsNormal( M, S ); </span>
true
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">F := FactorLoop( M, S ); </span>
&lt;loop of order 4&gt;
<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;,
function( x ) ... end )
<div class="example"><pre>
<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;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">IsNormal( M, S );</span>
true
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">F := FactorLoop( M, S );</span>
&lt;loop of order 4&gt;
<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;,
function( x ) ... end )
</pre></div>
<p><a id="X821F40748401D698" name="X821F40748401D698"></a></p>
@ -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><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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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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@ -444,19 +444,19 @@
<p>The following trivial example shows some of the implications and the naming conventions of <strong class="pkg">LOOPS</strong> at work:</p>
<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">L := LoopByCayleyTable( [ [ 1, 2 ], [ 2, 1 ] ] ); </span>
&lt;loop of order 2&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ IsLeftBolLoop( L ), L ] </span>
[ true, &lt;left Bol loop of order 2&gt; ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ HasIsLeftAlternativeLoop( L ), IsLeftAlternativeLoop( L ) ]; </span>
[ true, true ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ HasIsRightBolLoop( L ), IsRightBolLoop( L ) ]; </span>
[ false, true ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">L; </span>
&lt;Moufang loop of order 2&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ IsAssociative( L ), L ]; </span>
[ true, &lt;associative loop of order 2&gt; ]
<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">L := LoopByCayleyTable( [ [ 1, 2 ], [ 2, 1 ] ] );</span>
&lt;loop of order 2&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ IsLeftBolLoop( L ), L ]</span>
[ true, &lt;left Bol loop of order 2&gt; ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ HasIsLeftAlternativeLoop( L ), IsLeftAlternativeLoop( L ) ];</span>
[ true, true ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ HasIsRightBolLoop( L ), IsRightBolLoop( L ) ];</span>
[ false, true ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">L;</span>
&lt;Moufang loop of order 2&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ IsAssociative( L ), L ];</span>
[ true, &lt;associative loop of order 2&gt; ]
</pre></div>
<p>The analogous terminology for quasigroups of Bol-Moufang type is not standard yet, and hence is not supported in <strong class="pkg">LOOPS</strong> except for the situations explicitly noted above.</p>

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@ -202,13 +202,13 @@
<p>Returns: One loop (given as a section) whose multiplication group is equal to the transitive permutation group <var class="Arg">G</var>.</p>
<div class="example"><pre>
<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) ])
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">a:=AllLoopTablesInGroup(g,3,0);; Size(a); </span>
56
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">a:=AllLoopsWithMltGroup(g,3,0);; Size(a); </span>
52
<div class="example"><pre>
<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) ])
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">a:=AllLoopTablesInGroup(g,3,0);; Size(a);</span>
56
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">a:=AllLoopsWithMltGroup(g,3,0);; Size(a);</span>
52
</pre></div>

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@ -78,12 +78,34 @@
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
nle 64 and n∈{81,243}.
9.3-1 MoufangLoop
9.4-1 MoufangLoop
MoufangLoop( n, m )  function
Returns: The mth Moufang loop of order n in the library.
@ -107,20 +129,20 @@
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
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 [NV07] for a classification of small code loops.
9.4-1 CodeLoop
9.5-1 CodeLoop
CodeLoop( n, m )  function
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:
@ -141,13 +163,13 @@
Our labeling of Steiner loops of order 16 coincides with the labeling of
Steiner triple systems of order 15 in [CR99].
9.5-1 SteinerLoop
9.6-1 SteinerLoop
SteinerLoop( n, m )  function
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
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
RightConjugacyClosedLoop( n, m )  function
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
LeftConjugacyClosedLoop( n, m )  function
@ -188,8 +210,10 @@
Left conjugacy closed loops are obtained from right conjugacy closed loops
via Opposite.
The library named CC contains all nonassociative conjugacy closed loops of
order nle 27 and also of orders 2p and p^2 for all primes p.
The library named CC contains all CC loops of order 2le 2^kle 64, 3le 3^kle
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
nonassociative conjugacy closed loops of order p^2. Csörgő and Drápal [CD05]
@ -215,25 +239,25 @@
m + n ).
9.6-3 CCLoop and ConjugacyClosedLoop
9.7-3 CCLoop and ConjugacyClosedLoop
CCLoop( n, m )  function
ConjugacyClosedLoop( n, m )  function
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.
There are 5 and 107 such loops, respectively.
9.7-1 SmallLoop
9.8-1 SmallLoop
SmallLoop( n, m )  function
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],
there is precisely one Paige loop for every finite field.
@ -241,14 +265,14 @@
The library named Paige contains the smallest nonassociative simple Moufang
loop.
9.8-1 PaigeLoop
9.9-1 PaigeLoop
PaigeLoop( q )  function
Returns: The Paige loop constructed over the finite field of order q. Only
the case q=2 is implemented.
9.9 Nilpotent Loops
9.10 Nilpotent Loops
The library named nilpotent contains all nonassociative nilpotent loops of
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
nilpotent loops of order 22.
9.9-1 NilpotentLoop
9.10-1 NilpotentLoop
NilpotentLoop( n, m )  function
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
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 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).
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).
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
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 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
of order 96 and exponent 2.
9.11-1 InterestingLoop
9.12-1 InterestingLoop
InterestingLoop( n, m )  function
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
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
loops can be retrieved by the template ItpLibLoop(n,m).
9.12-1 ItpSmallLoop
9.13-1 ItpSmallLoop
ItpSmallLoop( n, m )  function
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#X8774304282654C58">9.2-2 RightBolLoop</a></span>
</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>
<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 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>
<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 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>
<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 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>
<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 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 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>
</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>
<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 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>
<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 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>
<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 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>
<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 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>
<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 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>
<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>
@ -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><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>
<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><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>
<p>Returns: The <var class="Arg">m</var>th Moufang loop of order <var class="Arg">n</var> in the library.</p>
@ -187,79 +215,79 @@
<p>The extent of the library is summarized below:</p>
<p class="center">\[
\begin{array}{r|rrrrrrrrrrrrrrrrrr}
order&amp;12&amp;16&amp;20&amp;24&amp;28&amp;32&amp;36&amp;40&amp;42&amp;44&amp;48&amp;52&amp;54&amp;56&amp;60&amp;64&amp;81&amp;243\cr
loops&amp;1 &amp;5 &amp;1 &amp;5 &amp;1 &amp;71&amp;4 &amp;5 &amp;1 &amp;1 &amp;51&amp;1 &amp;2 &amp;4 &amp;5 &amp;4262&amp; 5 &amp;72
\end{array}
<p class="center">\[
\begin{array}{r|rrrrrrrrrrrrrrrrrr}
order&amp;12&amp;16&amp;20&amp;24&amp;28&amp;32&amp;36&amp;40&amp;42&amp;44&amp;48&amp;52&amp;54&amp;56&amp;60&amp;64&amp;81&amp;243\cr
loops&amp;1 &amp;5 &amp;1 &amp;5 &amp;1 &amp;71&amp;4 &amp;5 &amp;1 &amp;1 &amp;51&amp;1 &amp;2 &amp;4 &amp;5 &amp;4262&amp; 5 &amp;72
\end{array}
\]</p>
<p>The <em>octonion loop</em> of order 16 (i.e., the multiplication loop of the basis elements in the 8-dimensional standard real octonion algebra) can be obtained as <code class="code">MoufangLoop(16,3)</code>.</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><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>
<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>
<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>
<div class="example"><pre>
<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.
It also contains the associative Steiner loops of order 4 and 8.
------
Extent of the library:
1 loop of order 4
1 loop of order 8
1 loop of order 10
2 loops of order 14
80 loops of order 16
true
<div class="example"><pre>
<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.
It also contains the associative Steiner loops of order 4 and 8.
------
Extent of the library:
1 loop of order 4
1 loop of order 8
1 loop of order 10
2 loops of order 14
80 loops of order 16
true
</pre></div>
<p>Our labeling of Steiner loops of order 16 coincides with the labeling of Steiner triple systems of order 15 in <a href="chapBib_mj.html#biBCoRo">[CR99]</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>
<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>
<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>Let <span class="SimpleMath">\(Q\)</span> be a right conjugacy closed loop, <span class="SimpleMath">\(G\)</span> its right multiplication group and <span class="SimpleMath">\(T\)</span> its right section. Then <span class="SimpleMath">\(\langle T\rangle = G\)</span> is a transitive group, and <span class="SimpleMath">\(T\)</span> is a union of conjugacy classes of <span class="SimpleMath">\(G\)</span>. Every right conjugacy closed loop of order <span class="SimpleMath">\(n\)</span> can therefore be represented as a union of certain conjugacy classes of a transitive group of degree <span class="SimpleMath">\(n\)</span>. This is how right conjugacy closed loops of order less than <span class="SimpleMath">\(28\)</span> are represented in <strong class="pkg">LOOPS</strong>. The following table summarizes the number of right conjugacy closed loops of a given order up to isomorphism:</p>
<p class="center">\[
\begin{array}{r|rrrrrrrrrrrrrrrr}
order &amp;6&amp; 8&amp;9&amp;10&amp; 12&amp;14&amp;15&amp; 16&amp; 18&amp; 20&amp;\cr
loops &amp;3&amp;19&amp;5&amp;16&amp;155&amp;97&amp; 17&amp;6317&amp;1901&amp;8248&amp;\cr
\hline
order &amp;21&amp; 22&amp; 24&amp; 25&amp; 26&amp; 27\cr
loops &amp;119&amp;10487&amp;471995&amp; 119&amp;151971&amp;152701
\end{array}
<p class="center">\[
\begin{array}{r|rrrrrrrrrrrrrrrr}
order &amp;6&amp; 8&amp;9&amp;10&amp; 12&amp;14&amp;15&amp; 16&amp; 18&amp; 20&amp;\cr
loops &amp;3&amp;19&amp;5&amp;16&amp;155&amp;97&amp; 17&amp;6317&amp;1901&amp;8248&amp;\cr
\hline
order &amp;21&amp; 22&amp; 24&amp; 25&amp; 26&amp; 27\cr
loops &amp;119&amp;10487&amp;471995&amp; 119&amp;151971&amp;152701
\end{array}
\]</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; 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>
<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; 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>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>
@ -295,7 +323,7 @@ true
<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; 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>
<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><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>
<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>
<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>
@ -324,14 +352,14 @@ true
<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>
<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>
<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>
@ -339,58 +367,60 @@ true
<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>
<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>
<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>
<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>
<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>
<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><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>
<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>
<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><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>
<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>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">SmallLoop( 6, 14 ); </span>
&lt;small loop 6/14&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">ItpSmallLoop( 6, 14 ); </span>
&lt;small loop 6/42&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LibraryLoop( "itp small", 6, 14 ); </span>
&lt;small loop 6/42&gt;
<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">SmallLoop( 6, 14 );</span>
&lt;small loop 6/14&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">ItpSmallLoop( 6, 14 );</span>
&lt;small loop 6/42&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LibraryLoop( "itp small", 6, 14 );</span>
&lt;small loop 6/42&gt;
</pre></div>
<p>Note that loops up to isotopism form a subset of the corresponding library of loops up to isomorphism. For instance, the above example shows that the 14th small loop of order 6 up to isotopism is in fact the 42nd small loop of order 6 up to isomorphism.</p>

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

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7
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@ -42,6 +42,9 @@
less than 64, Nova Science Publishers Inc., Commack, NY (1999), xviii+287
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
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,
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
order 64, European J. Combin., 27, 3 (2006), 444–460.

23
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@ -164,6 +164,18 @@
</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 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>,
@ -327,6 +339,17 @@
</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 class='BibEntry'>
[<span class='BibKeyLink'><a href="http://www.ams.org/mathscinet-getitem?mr=2206479">Voj06</a></span>] <b class='BibAuthor'>Vojtěchovský, P.</b>,

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

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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, 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 />
<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">AssociatedRightBruckLoop</code> <a href="chap8_mj.html#X8664CA927DD73DBE">8.1-1</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, middle <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 />
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 />
@ -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 />
<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">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 />
<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 />
@ -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 />
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 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 />
<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 />
@ -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, 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 />
<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 />
coset <a href="chap6_mj.html#X835F48248571364F">6.2-6</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 />
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">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 />
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 />
@ -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, 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">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">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 />
@ -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">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">IsomorphicCopyByNormalSubloop</code> <a href="chap6_mj.html#X8121DE3A78795040">6.11-7</a><br />
<code class="func">IsomorphicCopyByPerm</code> <a href="chap6_mj.html#X85B3E22679FD8D81">6.11-6</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">6.11-8</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">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">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">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, 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="chap4_mj.html#X7DE8405B82BC36A9">4.1</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 />
<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">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 />
@ -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, 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, 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, Steiner <a href="chap7_mj.html#X793600C9801F4F62">7.8-2</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="chap4_mj.html#X817132C887D3FD3A">4.9-2</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, 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 />
@ -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 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, 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="chap6_mj.html#X7D8E63A7824037CC">6.7-3</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">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">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">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 />
@ -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, dihedral <a href="chap8_mj.html#X7D7717C587BC2D1E">8.2-2</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, 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 />
@ -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 />
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 />
<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 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 />
@ -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 />
<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 />
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">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 />
@ -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">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 />
Paige loop <a href="chap9_mj.html#X8135C8FD8714C606">9.8</a><br />
<code class="func">PaigeLoop</code> <a href="chap9_mj.html#X7FCF4D6B7AD66D74">9.8-1</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">9.9-1</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">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">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">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">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">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 />
<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">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">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">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">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 />
section, left <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 />
<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 />
@ -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 />
<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">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 />
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 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 />
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 />

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@ -1,161 +0,0 @@
\begin{thebibliography}{DBGV12}
\bibitem[Art59]{Ar}
R.~Artzy.
\newblock On automorphic-inverse properties in loops.
\newblock {\em Proc. Amer. Math. Soc.}, 10:588{\textendash}591, 1959.
\bibitem[Art15]{Artic}
K.~Artic.
\newblock {\em On conjugacy closed loops and conjugacy closed loop folders}.
\newblock PhD thesis, RWTH Aachen University, 2015.
\bibitem[BP56]{BrPa}
R.~H. Bruck and L.~J. Paige.
\newblock Loops whose inner mappings are automorphisms.
\newblock {\em Ann. of Math. (2)}, 63:308{\textendash}323, 1956.
\bibitem[Bru58]{Br}
R.~H. Bruck.
\newblock {\em A survey of binary systems}.
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\bibitem[CD05]{CsDr}
P.~Cs{\"o}rg{\H o} and A.~Dr{\a'a}pal.
\newblock Left conjugacy closed loops of nilpotency class two.
\newblock {\em Results Math.}, 47(3-4):242{\textendash}265, 2005.
\bibitem[CR99]{CoRo}
C.~J. Colbourn and A.~Rosa.
\newblock {\em Triple systems}.
\newblock Oxford Mathematical Monographs. The Clarendon Press Oxford University
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\bibitem[DBGV12]{BaGrVo}
D.~A.~S. De~Barros, A.~Grishkov, and P.~Vojt{\v e}chovsk{\a'y}.
\newblock Commutative automorphic loops of order {$p^3$}.
\newblock {\em J. Algebra Appl.}, 11(5):1250100, 15, 2012.
\bibitem[Dr{\a'a}03]{DrapalCD}
A.~Dr{\a'a}pal.
\newblock Cyclic and dihedral constructions of even order.
\newblock {\em Comment. Math. Univ. Carolin.}, 44(4):593{\textendash}614, 2003.
\bibitem[DV06]{DrVo}
A.~Dr{\a'a}pal and P.~Vojt{\v e}chovsk{\a'y}.
\newblock Moufang loops that share associator and three quarters of their
multiplication tables.
\newblock {\em Rocky Mountain J. Math.}, 36(2):425{\textendash}455, 2006.
\bibitem[DV09]{DaVo}
D.~Daly and P.~Vojt{\v e}chovsk{\a'y}.
\newblock Enumeration of nilpotent loops via cohomology.
\newblock {\em J. Algebra}, 322(11):4080{\textendash}4098, 2009.
\bibitem[Fen69]{Fe}
F.~Fenyves.
\newblock Extra loops. {II}. {O}n loops with identities of {B}ol-{M}oufang
type.
\newblock {\em Publ. Math. Debrecen}, 16:187{\textendash}192, 1969.
\bibitem[GKN14]{GrKiNa}
A.~Grishkov, M.~Kinyon, and G.~P. Nagy.
\newblock Solvability of commutative automorphic loops.
\newblock {\em Proc. Amer. Math. Soc.}, 142(9):3029{\textendash}3037, 2014.
\bibitem[GMR99]{Go}
E.~G. Goodaire, S.~May, and M.~Raman.
\newblock {\em The {M}oufang loops of order less than 64}.
\newblock Nova Science Publishers Inc., Commack, NY, 1999.
\bibitem[JKNV11]{JoKiNaVo}
K.~W. Johnson, M.~K. Kinyon, G.~P. Nagy, and P.~Vojt{\v e}chovsk{\a'y}.
\newblock Searching for small simple automorphic loops.
\newblock {\em LMS J. Comput. Math.}, 14:200{\textendash}213, 2011.
\bibitem[JKV12]{JeKiVo}
P.~Jedli{\v c}ka, M.~Kinyon, and P.~Vojt{\v e}chovsk{\a'y}.
\newblock Nilpotency in automorphic loops of prime power order.
\newblock {\em J. Algebra}, 350:64{\textendash}76, 2012.
\bibitem[JM96]{JaMa}
M.~T. Jacobson and P.~Matthews.
\newblock Generating uniformly distributed random {L}atin squares.
\newblock {\em J. Combin. Des.}, 4(6):405{\textendash}437, 1996.
\bibitem[KKP02]{KiKuPh}
M.~K. Kinyon, K.~Kunen, and J.~D. Phillips.
\newblock Every diassociative {$A$}-loop is {M}oufang.
\newblock {\em Proc. Amer. Math. Soc.}, 130(3):619{\textendash}624, 2002.
\bibitem[KKPV16]{KiKuPhVo}
M.~K. Kinyon, K.~Kunen, J.~D. Phillips, and P.~Vojt{\v e}chovsk{\a'y}.
\newblock The structure of automorphic loops.
\newblock {\em Trans. Amer. Math. Soc.}, 368(12):8901{\textendash}8927, 2016.
\bibitem[KNV15]{KiNaVo2015}
M.~K. Kinyon, G.~P. Nagy, and P.~Vojt{\v e}chovsk{\a'y}.
\newblock Bol loops and bruck loops of order $pq$.
\newblock 2015.
\newblock preprint.
\bibitem[Kun00]{Kun}
K.~Kunen.
\newblock The structure of conjugacy closed loops.
\newblock {\em Trans. Amer. Math. Soc.}, 352(6):2889{\textendash}2911, 2000.
\bibitem[Lie87]{Li}
M.~W. Liebeck.
\newblock The classification of finite simple {M}oufang loops.
\newblock {\em Math. Proc. Cambridge Philos. Soc.}, 102(1):33{\textendash}47,
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\bibitem[Moo]{Mo}
G.~E. Moorhouse.
\newblock Bol loops of small order.
\newblock http://www.uwyo.edu/moorhouse/pub/bol/.
\bibitem[NV03]{NaVo2003}
G.~P. Nagy and P.~Vojt{\v e}chovsk{\a'y}.
\newblock Octonions, simple {M}oufang loops and triality.
\newblock {\em Quasigroups Related Systems}, 10:65{\textendash}94, 2003.
\bibitem[NV07]{NaVo2007}
G.~P. Nagy and P.~Vojt{\v e}chovsk{\a'y}.
\newblock The {M}oufang loops of order 64 and 81.
\newblock {\em J. Symbolic Comput.}, 42(9):871{\textendash}883, 2007.
\bibitem[Pfl90]{Pf}
H.~O. Pflugfelder.
\newblock {\em Quasigroups and loops: introduction}, volume~7 of {\em Sigma
Series in Pure Mathematics}.
\newblock Heldermann Verlag, Berlin, 1990.
\bibitem[PV05]{PhiVoj}
J.~D. Phillips and P.~Vojt{\v e}chovsk{\a'y}.
\newblock The varieties of loops of {B}ol-{M}oufang type.
\newblock {\em Algebra Universalis}, 54(3):259{\textendash}271, 2005.
\bibitem[SZ12]{SlZe2011}
M.~Slattery and A.~Zenisek.
\newblock Moufang loops of order 243.
\newblock {\em Commentationes Mathematicae Universitatis Carolinae},
53(3):423{\textendash}428, 2012.
\bibitem[Voj06]{Vo}
P.~Vojt{\v e}chovsk{\a'y}.
\newblock Toward the classification of {M}oufang loops of order 64.
\newblock {\em European J. Combin.}, 27(3):444{\textendash}460, 2006.
\bibitem[Voj15]{VoQRS}
P.~Vojt{\v e}chovsk{\a'y}.
\newblock Three lectures on automorphic loops.
\newblock {\em Quasigroups Related Systems}, 23(1):129{\textendash}163, 2015.
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R.~L. Wilson~Jr.
\newblock Quasidirect products of quasigroups.
\newblock {\em Comm. Algebra}, 3(9):835{\textendash}850, 1975.
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@ -260,6 +260,27 @@ Publishers, 1999.
MRNUMBER = {1689624 (2000a:20147)},
}
\bibitem{Greer}
Mark Greer.
\newblock{\it A class of loops categorically isomorphic to Bruck loops of odd order},
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}
Alexander Grishkov, Michael Kinyon and G\'abor Nagy.
\newblock {\it Solvability of commutative automorphic loops},
@ -591,6 +612,19 @@ preprint.
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}
Petr Vojt\v{e}chovsk\'y.
\newblock {\it Toward the classification of Moufang loops of order $64$},

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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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\backcite {DrapalCD}{{46}{8.2}{section.8.2}}
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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}
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\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}
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\indexentry{RightDivision@\texttt {RightDivision}|hyperpage}{23}
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\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}
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\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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\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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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 8.
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Chapter 9.
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[][]\T1/cmtt/m/n/10.95 LOOPS_my_library_data[2][k] \T1/ptm/m/n/10.95 is the num
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Appendix A.
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Appendix B.
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n/10.95 -\T1/cmtt/m/n/10.95 Loop\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 By\T1/ptm
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\BOOKMARK [0][-]{chapter.1}{Introduction}{}% 1
\BOOKMARK [1][-]{section.1.1}{License}{chapter.1}% 2
\BOOKMARK [1][-]{section.1.2}{Installation}{chapter.1}% 3
\BOOKMARK [1][-]{section.1.3}{Documentation}{chapter.1}% 4
\BOOKMARK [1][-]{section.1.4}{Test Files}{chapter.1}% 5
\BOOKMARK [1][-]{section.1.5}{Memory Management}{chapter.1}% 6
\BOOKMARK [1][-]{section.1.6}{Feedback}{chapter.1}% 7
\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
\BOOKMARK [1][-]{section.4.8}{Creating Quasigroups and Loops By Nuclear Extensions}{chapter.4}% 29
\BOOKMARK [1][-]{section.4.9}{Random Quasigroups and Loops}{chapter.4}% 30
\BOOKMARK [1][-]{section.4.10}{Conversions}{chapter.4}% 31
\BOOKMARK [1][-]{section.4.11}{Products of Quasigroups and Loops}{chapter.4}% 32
\BOOKMARK [1][-]{section.4.12}{Opposite Quasigroups and Loops}{chapter.4}% 33
\BOOKMARK [0][-]{chapter.5}{Basic Methods And Attributes}{}% 34
\BOOKMARK [1][-]{section.5.1}{Basic Attributes}{chapter.5}% 35
\BOOKMARK [1][-]{section.5.2}{Basic Arithmetic Operations}{chapter.5}% 36
\BOOKMARK [1][-]{section.5.3}{Powers and Inverses}{chapter.5}% 37
\BOOKMARK [1][-]{section.5.4}{Associators and Commutators}{chapter.5}% 38
\BOOKMARK [1][-]{section.5.5}{Generators}{chapter.5}% 39
\BOOKMARK [0][-]{chapter.6}{Methods Based on Permutation Groups}{}% 40
\BOOKMARK [1][-]{section.6.1}{Parent of a Quasigroup}{chapter.6}% 41
\BOOKMARK [1][-]{section.6.2}{Subquasigroups and Subloops}{chapter.6}% 42
\BOOKMARK [1][-]{section.6.3}{Translations and Sections}{chapter.6}% 43
\BOOKMARK [1][-]{section.6.4}{Multiplication Groups}{chapter.6}% 44
\BOOKMARK [1][-]{section.6.5}{Inner Mapping Groups}{chapter.6}% 45
\BOOKMARK [1][-]{section.6.6}{Nuclei, Commutant, Center, and Associator Subloop}{chapter.6}% 46
\BOOKMARK [1][-]{section.6.7}{Normal Subloops and Simple Loops}{chapter.6}% 47
\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
\BOOKMARK [0][-]{chapter.7}{Testing Properties of Quasigroups and Loops}{}% 53
\BOOKMARK [1][-]{section.7.1}{Associativity, Commutativity and Generalizations}{chapter.7}% 54
\BOOKMARK [1][-]{section.7.2}{Inverse Propeties}{chapter.7}% 55
\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
\BOOKMARK [1][-]{section.7.8}{Additonal Varieties of Loops}{chapter.7}% 61
\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
\BOOKMARK [0][-]{appendix.A}{Files}{}% 80
\BOOKMARK [0][-]{appendix.B}{Filters}{}% 81
\BOOKMARK [0][-]{appendix*.3}{References}{}% 82
\BOOKMARK [0][-]{section*.4}{Index}{}% 83

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doc/loops.tex
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\contentsline {chapter}{\numberline {1}\leavevmode {\color {Chapter }Introduction}}{6}{chapter.1}
\contentsline {section}{\numberline {1.1}\leavevmode {\color {Chapter }License}}{6}{section.1.1}
\contentsline {section}{\numberline {1.2}\leavevmode {\color {Chapter }Installation}}{6}{section.1.2}
\contentsline {section}{\numberline {1.3}\leavevmode {\color {Chapter }Documentation}}{6}{section.1.3}
\contentsline {section}{\numberline {1.4}\leavevmode {\color {Chapter }Test Files}}{7}{section.1.4}
\contentsline {section}{\numberline {1.5}\leavevmode {\color {Chapter }Memory Management}}{7}{section.1.5}
\contentsline {section}{\numberline {1.6}\leavevmode {\color {Chapter }Feedback}}{7}{section.1.6}
\contentsline {section}{\numberline {1.7}\leavevmode {\color {Chapter }Acknowledgment}}{7}{section.1.7}
\contentsline {chapter}{\numberline {2}\leavevmode {\color {Chapter }Mathematical Background}}{8}{chapter.2}
\contentsline {section}{\numberline {2.1}\leavevmode {\color {Chapter }Quasigroups and Loops}}{8}{section.2.1}
\contentsline {section}{\numberline {2.2}\leavevmode {\color {Chapter }Translations}}{8}{section.2.2}
\contentsline {section}{\numberline {2.3}\leavevmode {\color {Chapter }Subquasigroups and Subloops}}{9}{section.2.3}
\contentsline {section}{\numberline {2.4}\leavevmode {\color {Chapter }Nilpotence and Solvability}}{9}{section.2.4}
\contentsline {section}{\numberline {2.5}\leavevmode {\color {Chapter }Associators and Commutators}}{9}{section.2.5}
\contentsline {section}{\numberline {2.6}\leavevmode {\color {Chapter }Homomorphism and Homotopisms}}{9}{section.2.6}
\contentsline {chapter}{\numberline {3}\leavevmode {\color {Chapter }How the Package Works}}{11}{chapter.3}
\contentsline {section}{\numberline {3.1}\leavevmode {\color {Chapter }Representing Quasigroups}}{11}{section.3.1}
\contentsline {section}{\numberline {3.2}\leavevmode {\color {Chapter }Conversions between magmas, quasigroups, loops and groups}}{12}{section.3.2}
\contentsline {section}{\numberline {3.3}\leavevmode {\color {Chapter }Calculating with Quasigroups}}{12}{section.3.3}
\contentsline {section}{\numberline {3.4}\leavevmode {\color {Chapter }Naming, Viewing and Printing Quasigroups and their Elements}}{13}{section.3.4}
\contentsline {subsection}{\numberline {3.4.1}\leavevmode {\color {Chapter }SetQuasigroupElmName and SetLoopElmName}}{13}{subsection.3.4.1}
\contentsline {chapter}{\numberline {4}\leavevmode {\color {Chapter }Creating Quasigroups and Loops}}{14}{chapter.4}
\contentsline {section}{\numberline {4.1}\leavevmode {\color {Chapter }About Cayley Tables}}{14}{section.4.1}
\contentsline {section}{\numberline {4.2}\leavevmode {\color {Chapter }Testing Cayley Tables}}{14}{section.4.2}
\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}
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\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}
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\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}
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\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}
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\contentsline {section}{\numberline {4.11}\leavevmode {\color {Chapter }Products of Quasigroups and Loops}}{21}{section.4.11}
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\contentsline {section}{\numberline {4.12}\leavevmode {\color {Chapter }Opposite Quasigroups and Loops}}{21}{section.4.12}
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\contentsline {section}{\numberline {5.5}\leavevmode {\color {Chapter }Generators}}{24}{section.5.5}
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\contentsline {chapter}{\numberline {6}\leavevmode {\color {Chapter }Methods Based on Permutation Groups}}{26}{chapter.6}
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\contentsline {subsection}{\numberline {6.2.2}\leavevmode {\color {Chapter }Subloop}}{27}{subsection.6.2.2}
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\contentsline {subsection}{\numberline {6.2.5}\leavevmode {\color {Chapter }AllSubloops}}{28}{subsection.6.2.5}
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\contentsline {section}{\numberline {6.8}\leavevmode {\color {Chapter }Factor Loops}}{32}{section.6.8}
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\contentsline {section}{\numberline {6.9}\leavevmode {\color {Chapter }Nilpotency and Central Series}}{32}{section.6.9}
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\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}

64
doc/loops.xml
View File

@ -6,7 +6,8 @@
<!-- 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.)
@ -24,7 +25,7 @@
<TitlePage>
<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>
<Author>G&#225;bor P. Nagy
<Email>nagyg@math.u-szeged.hu</Email>
@ -34,7 +35,7 @@
<Email>petr@math.du.edu</Email>
<Address>Department of Mathematics, University of Denver</Address>
</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>
</TitlePage>
@ -65,7 +66,7 @@
<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.
@ -124,9 +125,9 @@ We welcome all comments and suggestions on <Package>LOOPS</Package>, especially
<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.
@ -260,8 +261,8 @@ DeclareCategory( "IsLoopElement",
DeclareRepresentation( "IsLoopElmRep",
IsPositionalObjectRep and IsMultiplicativeElementWithInverse, [1] );
## latin (auxiliary category for GAP to tell apart IsMagma and IsQuasigroup)
DeclareCategory( "IsLatin", IsObject );
DeclareCategory( "IsQuasigroup", IsMagma and IsLatin );
DeclareCategory( "IsLatinMagma", IsObject );
DeclareCategory( "IsQuasigroup", IsMagma and IsLatinMagma );
DeclareCategory( "IsLoop", IsQuasigroup and
IsMultiplicativeElementWithInverseCollection);
</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.
<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>
<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>
@ -1971,6 +1982,26 @@ The library named <Emph>left Bol</Emph> contains all nonassociative left Bol loo
</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 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>
</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:
<List>
@ -2141,7 +2172,9 @@ are 2623755 nilpotent loops of order 12, and 123794003928541545927226368 nilpote
<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>
<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>( IsRightAutomorphicLoop, 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>( IsRightAutomorphicLoop, IsLeftAutomorphicLoop and HasAntiautomorphicInverseProperty )</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 HasRightInverseProperty )</Code>
<Br/><Code>( IsMoufangLoop, IsAutomorphicLoop and HasWeakInverseProperty )</Code>
<Br/><Code>( IsMiddleAutomorphicLoop, IsCommutative )</Code>
<Br/><Code>( IsLeftAutomorphicLoop, IsLeftBruckLoop )</Code>
<Br/><Code>( IsLeftAutomorphicLoop, IsLCCLoop )</Code>
<Br/><Code>( IsRightAutomorphicLoop, IsRightBruckLoop )</Code>
<Br/><Code>( IsRightAutomorphicLoop, IsRCCLoop )</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>
@ -2368,4 +2402,4 @@ Many implications among properties of loops are built directly into <Package>LOO
<TheIndex/>
</Book>
</Book>

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doc/loops_bib.xml
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@ -1,559 +0,0 @@
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<name><first>Robert L.</first><last>Wilson Jr. </last></name>
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496
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@ -1,496 +0,0 @@
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title = {On conjugacy closed loops and conjugacy closed loop
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}
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author = {Moorhouse, G. E.},
title = {Bol loops of small order},
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author = {Pflugfelder, H. O.},
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address = {Berlin},
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}
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author = {Phillips, J. D. and Vojt{\v e}chovsk{\a'y}, P.},
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}
@article{ VoQRS,
author = {Vojt{\v e}chovsk{\a'y}, P.},
title = {Three lectures on automorphic loops},
journal = {Quasigroups Related Systems},
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number = {1},
year = {2015},
pages = {129{\textendash}163},
fjournal = {Quasigroups and Related Systems},
issn = {1561-2848},
mrclass = {20N05},
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mrreviewer = {{\a'A}gota Figula},
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@article{ Wi,
author = {Wilson Jr., R. L.},
title = {Quasidirect products of quasigroups},
journal = {Comm. Algebra},
volume = {3},
number = {9},
year = {1975},
pages = {835{\textendash}850},
fjournal = {Communications in Algebra},
issn = {0092-7872},
mrclass = {20N05},
mrnumber = {0376937 (51 \#13112)},
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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" ],
[ "distributive quasigroup left", "7.3-6", [ 7, 3, 6 ], 143, 38,
[ "distributive quasigroup left", "7.3-6", [ 7, 3, 6 ], 143, 39,
"distributive quasigroup left", "X7B76FD6E878ED4F1" ],
[ "quasigroup right distributive", "7.3-6", [ 7, 3, 6 ], 143, 38,
[ "quasigroup right distributive", "7.3-6", [ 7, 3, 6 ], 143, 39,
"quasigroup right distributive", "X7B76FD6E878ED4F1" ],
[ "distributive quasigroup right", "7.3-6", [ 7, 3, 6 ], 143, 38,
[ "distributive quasigroup right", "7.3-6", [ 7, 3, 6 ], 143, 39,
"distributive quasigroup right", "X7B76FD6E878ED4F1" ],
[ "quasigroup distributive", "7.3-6", [ 7, 3, 6 ], 143, 38,
[ "quasigroup distributive", "7.3-6", [ 7, 3, 6 ], 143, 39,
"quasigroup distributive", "X7B76FD6E878ED4F1" ],
[ "distributive quasigroup", "7.3-6", [ 7, 3, 6 ], 143, 38,
[ "distributive quasigroup", "7.3-6", [ 7, 3, 6 ], 143, 39,
"distributive quasigroup", "X7B76FD6E878ED4F1" ],
[ "\033[2XIsEntropic\033[102X", "7.3-7", [ 7, 3, 7 ], 160, 39,
[ "\033[2XIsEntropic\033[102X", "7.3-7", [ 7, 3, 7 ], 160, 40,
"isentropic", "X7F23D4D97A38D223" ],
[ "\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" ],
[ "quasigroup entropic", "7.3-7", [ 7, 3, 7 ], 160, 39,
[ "quasigroup entropic", "7.3-7", [ 7, 3, 7 ], 160, 40,
"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" ],
[ "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" ],
[ "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" ],
[ "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" ],
[ "alternative loop left", "7.4", [ 7, 4, 0 ], 170, 39,
[ "alternative loop left", "7.4", [ 7, 4, 0 ], 170, 40,
"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" ],
[ "alternative loop right", "7.4", [ 7, 4, 0 ], 170, 39,
[ "alternative loop right", "7.4", [ 7, 4, 0 ], 170, 40,
"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" ],
[ "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" ],
[ "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" ],
[ "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" ],
[ "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" ],
[ "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" ],
[ "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" ],
[ "flexible loop", "7.4", [ 7, 4, 0 ], 170, 39, "flexible loop",
[ "flexible loop", "7.4", [ 7, 4, 0 ], 170, 40, "flexible loop",
"X780D907986EBA6C7" ],
[ "loop flexible", "7.4", [ 7, 4, 0 ], 170, 39, "loop flexible",
[ "loop flexible", "7.4", [ 7, 4, 0 ], 170, 40, "loop flexible",
"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" ],
[ "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" ],
[ "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" ],
[ "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" ],
[ "LC loop", "7.4", [ 7, 4, 0 ], 170, 39, "lc loop", "X780D907986EBA6C7" ],
[ "loop LC", "7.4", [ 7, 4, 0 ], 170, 39, "loop lc", "X780D907986EBA6C7" ],
[ "RC loop", "7.4", [ 7, 4, 0 ], 170, 39, "rc loop", "X780D907986EBA6C7" ],
[ "loop RC", "7.4", [ 7, 4, 0 ], 170, 39, "loop rc", "X780D907986EBA6C7" ],
[ "Moufang loop", "7.4", [ 7, 4, 0 ], 170, 39, "moufang loop",
[ "LC loop", "7.4", [ 7, 4, 0 ], 170, 40, "lc loop", "X780D907986EBA6C7" ],
[ "loop LC", "7.4", [ 7, 4, 0 ], 170, 40, "loop lc", "X780D907986EBA6C7" ],
[ "RC loop", "7.4", [ 7, 4, 0 ], 170, 40, "rc loop", "X780D907986EBA6C7" ],
[ "loop RC", "7.4", [ 7, 4, 0 ], 170, 40, "loop rc", "X780D907986EBA6C7" ],
[ "Moufang loop", "7.4", [ 7, 4, 0 ], 170, 40, "moufang loop",
"X780D907986EBA6C7" ],
[ "loop Moufang", "7.4", [ 7, 4, 0 ], 170, 39, "loop moufang",
[ "loop Moufang", "7.4", [ 7, 4, 0 ], 170, 40, "loop moufang",
"X780D907986EBA6C7" ],
[ "C loop", "7.4", [ 7, 4, 0 ], 170, 39, "c loop", "X780D907986EBA6C7" ],
[ "loop C", "7.4", [ 7, 4, 0 ], 170, 39, "loop c", "X780D907986EBA6C7" ],
[ "extra loop", "7.4", [ 7, 4, 0 ], 170, 39, "extra loop",
[ "C loop", "7.4", [ 7, 4, 0 ], 170, 40, "c loop", "X780D907986EBA6C7" ],
[ "loop C", "7.4", [ 7, 4, 0 ], 170, 40, "loop c", "X780D907986EBA6C7" ],
[ "extra loop", "7.4", [ 7, 4, 0 ], 170, 40, "extra loop",
"X780D907986EBA6C7" ],
[ "loop extra", "7.4", [ 7, 4, 0 ], 170, 39, "loop extra",
[ "loop extra", "7.4", [ 7, 4, 0 ], 170, 40, "loop extra",
"X780D907986EBA6C7" ],
[ "alternative loop", "7.4", [ 7, 4, 0 ], 170, 39, "alternative loop",
[ "alternative loop", "7.4", [ 7, 4, 0 ], 170, 40, "alternative loop",
"X780D907986EBA6C7" ],
[ "loop alternative", "7.4", [ 7, 4, 0 ], 170, 39, "loop alternative",
[ "loop alternative", "7.4", [ 7, 4, 0 ], 170, 40, "loop alternative",
"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" ],
[ "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" ],
[ "\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" ],
[ "\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" ],
[ "\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" ],
[ "\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" ],
[ "\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" ],
[ "\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" ],
[ "\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" ],
[ "\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" ],
[ "\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,
40, "isrightnuclearsquareloop", "X807B3B21825E3076" ],
[ "\033[2XIsNuclearSquareLoop\033[102X", "7.4-11", [ 7, 4, 11 ], 273, 41,
41, "isrightnuclearsquareloop", "X807B3B21825E3076" ],
[ "\033[2XIsNuclearSquareLoop\033[102X", "7.4-11", [ 7, 4, 11 ], 273, 42,
"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" ],
[ "\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" ],
[ "\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" ],
[ "\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" ],
[ "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" ],
[ "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" ],
[ "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" ],
[ "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" ],
[ "power alternative loop", "7.5", [ 7, 5, 0 ], 324, 42,
[ "power alternative loop", "7.5", [ 7, 5, 0 ], 324, 43,
"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" ],
[ "\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" ],
[ "\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" ],
[ "\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" ],
[ "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" ],
[ "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" ],
[ "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" ],
[ "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" ],
[ "conjugacy closed loop", "7.6", [ 7, 6, 0 ], 346, 42,
[ "conjugacy closed loop", "7.6", [ 7, 6, 0 ], 346, 43,
"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" ],
[ "\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" ],
[ "\033[2XIsLeftConjugacyClosedLoop\033[102X", "7.6-1", [ 7, 6, 1 ], 358,
42, "isleftconjugacyclosedloop", "X784E08CD7B710AF4" ],
[ "\033[2XIsRCCLoop\033[102X", "7.6-2", [ 7, 6, 2 ], 364, 42, "isrccloop",
43, "isleftconjugacyclosedloop", "X784E08CD7B710AF4" ],
[ "\033[2XIsRCCLoop\033[102X", "7.6-2", [ 7, 6, 2 ], 364, 43, "isrccloop",
"X7B3016B47A1A8213" ],
[ "\033[2XIsRightConjugacyClosedLoop\033[102X", "7.6-2", [ 7, 6, 2 ], 364,
42, "isrightconjugacyclosedloop", "X7B3016B47A1A8213" ],
[ "\033[2XIsCCLoop\033[102X", "7.6-3", [ 7, 6, 3 ], 370, 42, "isccloop",
43, "isrightconjugacyclosedloop", "X7B3016B47A1A8213" ],
[ "\033[2XIsCCLoop\033[102X", "7.6-3", [ 7, 6, 3 ], 370, 43, "isccloop",
"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" ],
[ "\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" ],
[ "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" ],
[ "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" ],
[ "automorphic loop left", "7.7", [ 7, 7, 0 ], 384, 43,
[ "automorphic loop left", "7.7", [ 7, 7, 0 ], 384, 44,
"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" ],
[ "automorphic loop middle", "7.7", [ 7, 7, 0 ], 384, 43,
[ "automorphic loop middle", "7.7", [ 7, 7, 0 ], 384, 44,
"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" ],
[ "automorphic loop right", "7.7", [ 7, 7, 0 ], 384, 43,
[ "automorphic loop right", "7.7", [ 7, 7, 0 ], 384, 44,
"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" ],
[ "automorphic loop", "7.7", [ 7, 7, 0 ], 384, 43, "automorphic loop",
[ "automorphic loop", "7.7", [ 7, 7, 0 ], 384, 44, "automorphic loop",
"X793B22EA8643C667" ],
[ "loop automorphic", "7.7", [ 7, 7, 0 ], 384, 43, "loop automorphic",
[ "loop automorphic", "7.7", [ 7, 7, 0 ], 384, 44, "loop automorphic",
"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" ],
[ "\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" ],
[ "\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" ],
[ "\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" ],
[ "\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" ],
[ "\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" ],
[ "\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" ],
[ "\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" ],
[ "\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" ],
[ "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" ],
[ "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" ],
[ "\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" ],
[ "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" ],
[ "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" ],
[ "\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" ],
[ "\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" ],
[ "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" ],
[ "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" ],
[ "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" ],
[ "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" ],
[ "\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" ],
[ "\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" ],
[ "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" ],
[ "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" ],
[ "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" ],
[ "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" ],
[ "\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" ],
[ "\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" ],
[ "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" ],
[ "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" ],
[ "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" ],
[ "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" ],
[ "\033[2XIsExactGroupFactorization\033[102X", "8.1-2", [ 8, 1, 2 ], 26,
45, "isexactgroupfactorization", "X82FC16F386CE11F1" ],
[ "exact group factorization", "8.1-2", [ 8, 1, 2 ], 26, 45,
46, "isexactgroupfactorization", "X82FC16F386CE11F1" ],
[ "exact group factorization", "8.1-2", [ 8, 1, 2 ], 26, 46,
"exact group factorization", "X82FC16F386CE11F1" ],
[ "\033[2XRightBolLoopByExactGroupFactorization\033[102X", "8.1-3",
[ 8, 1, 3 ], 35, 45, "rightbolloopbyexactgroupfactorization",
[ 8, 1, 3 ], 35, 46, "rightbolloopbyexactgroupfactorization",
"X7DCA64807F899127" ],
[ "modification Moufang", "8.2", [ 8, 2, 0 ], 47, 46,
[ "modification Moufang", "8.2", [ 8, 2, 0 ], 47, 47,
"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" ],
[ "modification cyclic", "8.2-1", [ 8, 2, 1 ], 57, 46,
[ "modification cyclic", "8.2-1", [ 8, 2, 1 ], 57, 47,
"modification cyclic", "X7B3165C083709831" ],
[ "\033[2XLoopByDihedralModification\033[102X", "8.2-2", [ 8, 2, 2 ], 70,
46, "loopbydihedralmodification", "X7D7717C587BC2D1E" ],
[ "modification dihedral", "8.2-2", [ 8, 2, 2 ], 70, 46,
47, "loopbydihedralmodification", "X7D7717C587BC2D1E" ],
[ "modification dihedral", "8.2-2", [ 8, 2, 2 ], 70, 47,
"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" ],
[ "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" ],
[ "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" ],
[ "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" ],
[ "\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" ],
[ "\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" ],
[ "\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" ],
[ "\033[2XAllProperLoopTablesInGroup\033[102X", "8.4-2", [ 8, 4, 2 ], 152,
47, "allproperlooptablesingroup", "X7854C8E382DC8E8B" ],
[ "\033[2XOneLoopTableInGroup\033[102X", "8.4-3", [ 8, 4, 3 ], 158, 47,
48, "allproperlooptablesingroup", "X7854C8E382DC8E8B" ],
[ "\033[2XOneLoopTableInGroup\033[102X", "8.4-3", [ 8, 4, 3 ], 158, 48,
"onelooptableingroup", "X7BFFC66A824BA6AA" ],
[ "\033[2XOneProperLoopTableInGroup\033[102X", "8.4-4", [ 8, 4, 4 ], 164,
48, "oneproperlooptableingroup", "X84C5A76585B335FF" ],
[ "\033[2XAllLoopsWithMltGroup\033[102X", "8.4-5", [ 8, 4, 5 ], 170, 48,
49, "oneproperlooptableingroup", "X84C5A76585B335FF" ],
[ "\033[2XAllLoopsWithMltGroup\033[102X", "8.4-5", [ 8, 4, 5 ], 170, 49,
"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" ],
[ "\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" ],
[ "\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" ],
[ "\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" ],
[ "\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" ],
[ "\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" ],
[ "\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" ],
[ "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" ],
[ "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" ],
[ "\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" ],
[ "\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" ],
[ "\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" ],
[ "\033[2XRightConjugacyClosedLoop\033[102X", "9.6-1", [ 9, 6, 1 ], 173,
52, "rightconjugacyclosedloop", "X806B2DE67990E42F" ],
[ "\033[2XLCCLoop\033[102X", "9.6-2", [ 9, 6, 2 ], 180, 52, "lccloop",
[ "\033[2XRightConjugacyClosedLoop\033[102X", "9.7-1", [ 9, 7, 1 ], 195,
53, "rightconjugacyclosedloop", "X806B2DE67990E42F" ],
[ "\033[2XLCCLoop\033[102X", "9.7-2", [ 9, 7, 2 ], 202, 53, "lccloop",
"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" ],
[ "\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" ],
[ "\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" ],
[ "\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" ],
[ "Paige loop", "9.8", [ 9, 8, 0 ], 235, 53, "paige loop",
[ "Paige loop", "9.9", [ 9, 9, 0 ], 259, 54, "paige loop",
"X8135C8FD8714C606" ],
[ "loop Paige", "9.8", [ 9, 8, 0 ], 235, 53, "loop paige",
[ "loop Paige", "9.9", [ 9, 9, 0 ], 259, 54, "loop paige",
"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" ],
[ "\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" ],
[ "\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" ],
[ "sedenion loop", "9.11", [ 9, 11, 0 ], 283, 54, "sedenion loop",
[ "sedenion loop", "9.12", [ 9, 12, 0 ], 309, 55, "sedenion loop",
"X843BD73F788049F7" ],
[ "loop sedenion", "9.11", [ 9, 11, 0 ], 283, 54, "loop sedenion",
[ "loop sedenion", "9.12", [ 9, 12, 0 ], 309, 55, "loop sedenion",
"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" ],
[ "\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" ] ]
);

6
etc/gapdoc.txt
View File

@ -22,7 +22,7 @@ chooser.html
When files are ready, run the following in GAP:
# 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";;
files := [];;
bookname := "loops";;
@ -52,7 +52,7 @@ GAPDoc2HTMLPrintHTMLFiles(h, path);
# h := GAPDoc2HTML(r, path );;
# GAPDoc2HTMLPrintHTMLFiles(h, path);
# now produce .ps, .dvi from .tex,
# and copy loops.* as manual.* for extensions pdf, ps, dvi
# now produce .ps from .tex
# and copy loops.* as manual.* for extensions pdf, ps
# delete auxiliary files

4
gap/banner.g
View File

@ -2,7 +2,7 @@
##
#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 P. Vojtechovsky (University of Denver, USA)
@ -12,7 +12,7 @@ if not QUIET and BANNER then
Print(
" ______________________________________________________\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",
" nagyg@math.u-szeged.hu petr@math.du.edu \n",
" ------------------------------------------------------\n",

12
gap/classes.gi
View File

@ -2,7 +2,7 @@
##
#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 P. Vojtechovsky (University of Denver, USA)
@ -887,16 +887,14 @@ end);
InstallMethod( IsALoop, "for loop",
[ IsLoop ],
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);
# implies
InstallTrueMethod( IsLeftALoop, IsALoop );
InstallTrueMethod( IsRightALoop, IsALoop );
InstallTrueMethod( IsMiddleALoop, IsALoop );
InstallTrueMethod( IsMiddleALoop, IsCommutative );
InstallTrueMethod( IsALoop, IsLeftALoop and IsCommutative );
InstallTrueMethod( IsALoop, IsRightALoop and IsCommutative );
InstallTrueMethod( IsLeftALoop, IsRightALoop and HasAntiautomorphicInverseProperty );
InstallTrueMethod( IsRightALoop, IsLeftALoop and HasAntiautomorphicInverseProperty );
InstallTrueMethod( IsFlexible, IsMiddleALoop );
@ -909,8 +907,12 @@ InstallTrueMethod( IsMoufangLoop, IsALoop and HasRightInverseProperty );
InstallTrueMethod( IsMoufangLoop, IsALoop and HasWeakInverseProperty );
# is implied by
InstallTrueMethod( IsMiddleALoop, IsCommutative );
InstallTrueMethod( IsLeftALoop, IsLeftBruckLoop );
InstallTrueMethod( IsLeftALoop, IsLCCLoop );
InstallTrueMethod( IsRightALoop, IsRightBruckLoop );
InstallTrueMethod( IsRightALoop, IsRCCLoop );
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]
##
#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 P. Vojtechovsky (University of Denver, USA)
@ -36,6 +36,8 @@ DeclareGlobalFunction( "SmallLoop" );
DeclareGlobalFunction( "InterestingLoop" );
DeclareGlobalFunction( "NilpotentLoop" );
DeclareGlobalFunction( "AutomorphicLoop" );
DeclareGlobalFunction( "LeftBruckLoop" );
DeclareGlobalFunction( "RightBruckLoop" );
# up to isotopism
@ -52,3 +54,4 @@ DeclareGlobalFunction( "LOOPS_ActivateRCCLoop" );
DeclareGlobalFunction( "LOOPS_ActivateCCLoop" );
DeclareGlobalFunction( "LOOPS_ActivateNilpotentLoop" );
DeclareGlobalFunction( "LOOPS_ActivateAutomorphicLoop" );
DeclareGlobalFunction( "LOOPS_ActivateRightBruckLoop" );

213
gap/examples.gi
View File

@ -2,7 +2,7 @@
##
#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 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/nilpotent.tbl"); # nilpotent loops
ReadPackage("loops", "data/automorphic.tbl"); # automorphic loops
ReadPackage("loops", "data/rightbruck.tbl"); # right Bruck 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 = "nilpotent" then return LOOPS_nilpotent_data;
elif name = "automorphic" then return LOOPS_automorphic_data;
elif name = "right Bruck" then return LOOPS_right_bruck_data;
#up to isotopism
elif name = "itp small" then return LOOPS_itp_small_data;
fi;
@ -74,11 +76,8 @@ end);
InstallGlobalFunction( DisplayLibraryInfo, function( name )
local s, lib, k;
# up to isomorphism
if name = "left 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.";
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
if name = "left Bol" or name = "right Bol" then
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 = "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.";
elif name = "Paige" then
@ -88,12 +87,9 @@ InstallGlobalFunction( DisplayLibraryInfo, function( name )
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.";
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.";
elif name = "RCC" then
s := "The library contains all nonassociative RCC 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
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" or name = "LCC" then
s := Concatenation( "The library contains all nonassociative ", name, " loops of order less than 28." );
elif name = "small" then
s := "The library contains all nonassociative loops of order less than 7.";
elif name = "interesting" then
@ -103,23 +99,27 @@ InstallGlobalFunction( DisplayLibraryInfo, function( name )
elif name = "automorphic" then
s := "The library contains:\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 243 that are central\n");
s := Concatenation(s," extensions of Z_3 by F, where F is not the elem. ab. 3-group.\n");
s := Concatenation(s,"Note: Abelian groups are included among the commutative loops.");
s := Concatenation(s," - all commutative automorphic loops of order 3, 9, 27, 81.");
elif name = "left Bruck" or name = "right Bruck" then
s := Concatenation( "The library contains all ", name, " loops of orders 3, 9, 27 and 81." );
# up to isotopism
elif name = "itp small" then
s := "The library contains all nonassociative loops of order less than 7 up to isotopism.";
else
Info( InfoWarning, 1, Concatenation(
"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;
fi;
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 );
for k in [1..Length( lib[ 1 ] ) ] do
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 ] ) );
fi;
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)/2 loops of order p*q for primes p>q>2 such that q divides p+1" );
fi;
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;
s := Concatenation( s, "\n" );
Print( s );
@ -436,7 +436,48 @@ end);
InstallGlobalFunction( LOOPS_ActivateCCLoop,
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
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,
function( n, m )
local i, pos_n, factor_id, F, dim, coords, basis, coc;
if IsEmpty( LOOPS_automorphic_cocycles ) then # only read on demand
ReadPackage( "loops", "data/automorphic/automorphic_cocycles.tbl");
# decode cocycles
for i in [1..3] do
LOOPS_automorphic_cocycles[ i ] := List( LOOPS_automorphic_cocycles[ i ],
c -> LOOPS_DecodeCocycle( [ 3^(i+2), true, c ], [0,1,2] )
);
od;
# separate coordinates (from a long string )
for i in [1..3] do
LOOPS_automorphic_coordinates[ i ] := SplitString( LOOPS_automorphic_coordinates[ i ], " " );
od;
fi;
# returns the associated Gamma loop (which here always happens to be automorphic)
# improve later
local P, L, s, Ls, ct, i, j, pos, f;
P := LeftBruckLoop( n, m );
L := LeftMultiplicationGroup( P );;
s := List(Elements(L), x -> x^2 );;
Ls := List([1..n], i -> LeftTranslation( P, Elements(P)[i] ) );;
ct := List([1..n],i->0*[1..n]);;
for i in [1..n] do for j in [1..n] do
pos := Position( s, Ls[i]*Ls[j]*Ls[i]^(-1)*Ls[j]^(-1) );
f := Elements(L)[pos];
ct[i][j] := 1^(f*Ls[j]*Ls[i]);
od; od;
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
pos_n := Position( [27,81,243], n );
factor_id := LOOPS_CharToDigit( LOOPS_automorphic_coordinates[ pos_n ][ m ][ 1 ] );
F := AutomorphicLoop( n/3, factor_id );
pos_n := Position( [27,81], n );
factor_id := LOOPS_CharToDigit( LOOPS_right_bruck_coordinates[ pos_n ][ m ][ 1 ] );
F := RightBruckLoop( n/3, factor_id );
# basis (only decode cocycles at first usage)
if IsString( LOOPS_right_bruck_cocycles[ pos_n ][ 1 ][ 3 ] ) then # not converted yet
LOOPS_right_bruck_cocycles[ pos_n ] := List( LOOPS_right_bruck_cocycles[ pos_n ],
coc -> LOOPS_DecodeCocycle( coc, [0,1,2] )
);
fi;
basis := LOOPS_right_bruck_cocycles[ pos_n ];
# coordinates determining the cocycle
dim := Length( LOOPS_automorphic_bases[ pos_n ][ factor_id ] );
coords := LOOPS_automorphic_coordinates[ pos_n ][ m ];
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, dim );
coords := LOOPS_ConvertBase( coords, 91, 3, Length( basis ) );
coords := List( coords, LOOPS_CharToDigit );
# basis
basis := List( LOOPS_automorphic_bases[ pos_n ][ factor_id ],
i -> LOOPS_automorphic_cocycles[ pos_n ][ i ]
);
# calculate cocycle
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 LoopByExtension( AutomorphicLoop(3,1), F, List([1..n/3], i -> () ), coc );
return LoopByExtension( RightBruckLoop(3,1), F, List([1..n/3], i -> () ), coc );
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;
# 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 );
fi;
lib := LOOPS_LibraryByName( name );
# extent of the library
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 )
pos_n := fail;
case := false;
if name="left Bol" or name="right Bol" then
if name="left Bol" then
divs := DivisorsInt( n );
if Length( divs ) = 4 and not IsInt( divs[3]/divs[2] ) then # case n = p*q
q := divs[ 2 ];
@ -633,13 +681,13 @@ InstallGlobalFunction( LibraryLoop, function( name, n, m )
fi;
if name="CC" then
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 ];
case := [p,"p^2"];
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.");
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 ];
case := [p,"2*p"];
if not divs[2] = 2 then
@ -670,9 +718,6 @@ InstallGlobalFunction( LibraryLoop, function( name, n, m )
if name = "left Bol" then
loop := LOOPS_ActivateLeftBolLoop( pos_n, m, case );
SetIsLeftBolLoop( loop, true );
elif name = "right Bol" then
loop := OppositeLoop( LOOPS_ActivateLeftBolLoop( pos_n, m, case ) );
SetIsRightBolLoop( loop, true );
elif name = "Moufang" then
# renaming loops so that they agree with Goodaire's classification
PG := List([1..243], i->());
@ -701,14 +746,15 @@ InstallGlobalFunction( LibraryLoop, function( name, n, m )
loop := LOOPS_ActivateSteinerLoop( n, pos_n, m );
SetIsSteinerLoop( loop, true );
elif name = "CC" then
loop := LOOPS_ActivateCCLoop( n, pos_n, m, case );
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 );
fi;
SetIsCCLoop( loop, true );
elif name = "RCC" then
loop := LOOPS_ActivateRCCLoop( n, pos_n, m );
SetIsRCCLoop( loop, true );
elif name = "LCC" then
loop := OppositeLoop( LOOPS_ActivateRCCLoop( n, pos_n, m ) );
SetIsLCCLoop( loop, true );
elif name = "small" then
loop := LoopByCayleyTable( LOOPS_DecodeCayleyTable( lib[ 3 ][ pos_n ][ m ] ) );
elif name = "interesting" then
@ -725,12 +771,19 @@ InstallGlobalFunction( LibraryLoop, function( name, n, m )
elif name = "nilpotent" then
loop := LOOPS_ActivateNilpotentLoop( lib[ 3 ][ pos_n ][ m ] );
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 ] ) );
else # use cocycles
loop := LOOPS_ActivateAutomorphicLoop( n, m );
else # use associated left Bruck loop
loop := LOOPS_ActivateAutomorphicLoop( n, m );
fi;
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
elif name = "itp small" then
return LibraryLoop( "small", n, lib[ 3 ][ pos_n ][ m ] );
@ -762,6 +815,8 @@ end);
#F InterestingLoop( n, m )
#F NilpotentLoop( n, m )
#F AutomorphicLoop( n, m )
#F LeftBruckLoop( n, m )
#F RightBruckLoop( n, m )
#F ItpSmallLoop( n, m )
##
@ -770,7 +825,11 @@ InstallGlobalFunction( LeftBolLoop, function( n, m )
end);
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);
InstallGlobalFunction( MoufangLoop, function( n, m )
@ -808,11 +867,15 @@ InstallGlobalFunction( RightConjugacyClosedLoop, function( n, m )
end);
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);
InstallGlobalFunction( LeftConjugacyClosedLoop, function( n, m )
return LibraryLoop( "LCC", n, m );
return LCCLoop( n, m );
end);
InstallGlobalFunction( SmallLoop, function( n, m )
@ -831,6 +894,18 @@ InstallGlobalFunction( AutomorphicLoop, function( n, m )
return LibraryLoop( "automorphic", n, m );
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 )
return LibraryLoop( "itp small", n, m );
end);

4
gap/iso.gd
View File

@ -2,7 +2,7 @@
##
#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 P. Vojtechovsky (University of Denver, USA)
@ -23,6 +23,8 @@ DeclareOperation( "IsomorphismQuasigroups", [ IsQuasigroup, IsQuasigroup ] );
DeclareOperation( "IsomorphismLoops", [ IsLoop, IsLoop ] );
DeclareOperation( "QuasigroupsUpToIsomorphism", [ IsList ] );
DeclareOperation( "LoopsUpToIsomorphism", [ IsList ] );
DeclareOperation( "QuasigroupIsomorph", [ IsQuasigroup, IsPerm ] );
DeclareOperation( "LoopIsomorph", [ IsLoop, IsPerm ] );
DeclareOperation( "IsomorphicCopyByPerm", [ IsQuasigroup, IsPerm ] );
DeclareOperation( "IsomorphicCopyByNormalSubloop", [ IsLoop, IsLoop ] );

65
gap/iso.gi
View File

@ -2,7 +2,7 @@
##
#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 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
## 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 ],
function( Q, p )
local ctQ, ct, inv_p;
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 );
ct := List([1..Size(Q)], i-> List([1..Size(Q)], j ->
( ctQ[ i^inv_p ][ j^inv_p ] )^p
) );
if IsLoop( Q ) then return LoopByCayleyTable( ct ); fi;
return QuasigroupByCayleyTable( ct );
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 )
@ -594,15 +618,13 @@ end);
##
## If L1, L2 are isotopic loops, returns true, else fail.
# (MATH) First we calculate all principal loop isotopes of L1 of the form
# PrincipalLoopIsotope(L1, f, g), where f, g, are elements 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.
# (MATH) We check for isomorphism of L2 against all principal
# isotopes of L1.
InstallMethod( IsotopismLoops, "for two loops",
[ IsLoop, IsLoop ],
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
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;
# now trying to construct an isotopism
istps := [];
fg := [];
for f in L1 do for g in L1 do
Add(istps, PrincipalLoopIsotope( L1, f, g ));
Add(fg, [ f, g ] );
od; od;
for L in LoopsUpToIsomorphism( istps ) do
L := PrincipalLoopIsotope( L1, f, g );
phi := IsomorphismLoops( L, L2 );
if not phi = fail then
# 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 );
beta := LeftTranslation( L1, f );
# we also applied an isomorphism (1,f*g) inside PrincipalLoopIsotope
@ -649,7 +662,7 @@ function( L1, L2 )
gamma := gamma * phi;
return [ alpha, beta, gamma ];
fi;
od;
od; od;
return fail;
end);

13
gap/memory.gi
View File

@ -2,7 +2,7 @@
##
#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 P. Vojtechovsky (University of Denver, USA)
@ -21,9 +21,14 @@ InstallGlobalFunction( LOOPS_FreeMemory, function( )
LOOPS_rcc_transitive_groups := [];
LOOPS_rcc_sections := List( [1..Length(LOOPS_rcc_data[1])], i-> [] );
LOOPS_rcc_conjugacy_classes := [ [], [] ];
# automorphic loops
LOOPS_automorphic_cocycles := [];
LOOPS_automorphic_coordinates := [];
# cc loops
LOOPS_cc_used_factors := [];
LOOPS_cc_cocycles := [];
LOOPS_cc_bases := [];
LOOPS_cc_coordinates := [];
# right Bruck loops
LOOPS_right_bruck_cocycles := [];
LOOPS_right_bruck_coordinates := [];
GASMAN("collect");
return GasmanStatistics().full.deadkb;
end);

297
gap/quasigroups.gd
View File

@ -1,151 +1,146 @@
#############################################################################
##
#W quasigroups.gd Representing, creating and displaying quasigroups [loops]
##
#H @(#)$Id: quasigroups.gd, v 3.2.0 2016/05/02 gap Exp $
##
#Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary),
#Y P. Vojtechovsky (University of Denver, USA)
##
#############################################################################
## GAP CATEGORIES AND REPRESENTATIONS
## -------------------------------------------------------------------------
## Categories convenient for defining quasigroups
## element which is an admissible argument for the right argument of /
DeclareCategory( "IsRightQuotientElement", IsExtLElement);
DeclareCategoryCollections("IsRightQuotientElement");
DeclareCategoryCollections("IsRightQuotientElementCollection");
## Every element with an inverse can form right quotients
## (in fact, in some sense it might be enough to have just a left inverse,
## but there doesn't seem to be any benefit to delving to that level of
## detail at this point.)
InstallTrueMethod(IsRightQuotientElement, IsMultiplicativeElementWithInverse);
## Now what we would like to do is re-declare
## DeclareOperation( "/", [IsExtRElement, IsRightQuotientElement] );
## but we can't since "/" is in the kernel, so we will have to content
## ourselves with InstallOtherMethod() calls on /. (I am not actually sure what
## the practical upshot of that is, i.e. if it has any shortcomings as compared
## to if we could declare "/" more generally.)
## Element which is admissible for the left argument of LeftQuotient()
DeclareCategory( "IsLeftQuotientElement", IsExtRElement);
DeclareCategoryCollections("IsLeftQuotientElement");
DeclareCategoryCollections("IsLeftQuotientElementCollection");
## Every element with an inverse can form left quotients
InstallTrueMethod(IsLeftQuotientElement, IsMultiplicativeElementWithInverse);
## Again, ideally (in some sense) we'd like to redeclare
## DeclareOperation("LeftQuotient", [IsLeftQuotientElement,IsExtLElement]);
## element of a quasigroup
DeclareSynonym( "IsQuasigroupElement",
IsMultiplicativeElement and
IsLeftQuotientElement and IsRightQuotientElement );
DeclareRepresentation( "IsQuasigroupElmRep",
IsPositionalObjectRep and IsMultiplicativeElement, [1] );
## element of a loop
DeclareSynonym( "IsLoopElement",
IsQuasigroupElement and IsMultiplicativeElementWithInverse );
DeclareRepresentation( "IsLoopElmRep",
IsPositionalObjectRep and IsMultiplicativeElementWithInverse, [1] );
## Right quasigroup
DeclareCategory("IsRightQuasigroup",
IsMagma and IsRightQuotientElementCollection);
## Left quasigroup
DeclareCategory("IsLeftQuasigroup",
IsMagma and IsLeftQuotientElementCollection);
## quasigroup
DeclareSynonym( "IsQuasigroup", IsRightQuasigroup and IsLeftQuasigroup );
## loop
DeclareSynonym( "IsLoop", IsQuasigroup and IsMagmaWithOne and
IsMultiplicativeElementWithInverseCollection);
#############################################################################
## TESTING MULTIPLICATION TABLES
## -------------------------------------------------------------------------
DeclareProperty( "IsLeftQuasigroupTable", IsMatrix );
DeclareProperty( "IsRightQuasigroupTable", IsMatrix );
DeclareSynonym( "IsQuasigroupTable",
IsLeftQuasigroupTable and IsRightQuasigroupTable );
DeclareSynonym( "IsQuasigroupCayleyTable", IsQuasigroupTable );
DeclareProperty( "IsLoopTable", IsMatrix );
DeclareSynonym( "IsLoopCayleyTable", IsLoopTable );
DeclareGlobalFunction("CanonicalCayleyTableOfLeftQuasigroupTable");
DeclareOperation( "CanonicalCayleyTable", [ IsMatrix ] );
DeclareOperation( "NormalizedQuasigroupTable", [ IsMatrix ] );
#############################################################################
## CREATING QUASIGROUPS AND LOOPS MANUALLY
## -------------------------------------------------------------------------
DeclareAttribute( "CayleyTable", IsMagma );
DeclareOperation( "QuasigroupByCayleyTable", [ IsMatrix ] );
DeclareOperation( "LoopByCayleyTable", [ IsMatrix ] );
DeclareOperation( "SpecifyElmNamePrefix", [ IsCollection, IsString ] );
DeclareSynonym( "SetQuasigroupElmName", SpecifyElmNamePrefix );
DeclareSynonym( "SetLoopElmName", SpecifyElmNamePrefix );
DeclareOperation( "BindElmNames", [ IsMagma ] );
DeclareAttribute( "ConstructorFromTable", IsMagma );
DeclareOperation( "CanonicalCopy", [ IsMagma ] );
#############################################################################
## CREATING QUASIGROUPS AND LOOPS FROM A FILE
## -------------------------------------------------------------------------
DeclareOperation( "QuasigroupFromFile", [ IsString, IsString ] );
DeclareOperation( "LoopFromFile", [ IsString, IsString ] );
#############################################################################
## CREATING QUASIGROUPS AND LOOPS BY SECTIONS
## -------------------------------------------------------------------------
DeclareGlobalFunction("CayleyTableByPerms");
DeclareOperation( "QuasigroupByLeftSection", [ IsPermCollection ] );
DeclareOperation( "LoopByLeftSection", [ IsPermCollection ] );
DeclareOperation( "QuasigroupByRightSection", [ IsPermCollection ] );
DeclareOperation( "LoopByRightSection", [ IsPermCollection ] );
DeclareOperation( "QuasigroupByRightFolder", [ IsGroup, IsGroup, IsMultiplicativeElementCollection ] );
DeclareOperation( "LoopByRightFolder", [ IsGroup, IsGroup, IsMultiplicativeElementCollection ] );
#############################################################################
## CONVERSIONS
## -------------------------------------------------------------------------
DeclareOperation( "IntoQuasigroup", [ IsMagma ] );
DeclareOperation( "PrincipalLoopIsotope",
[ IsQuasigroup, IsQuasigroupElement, IsQuasigroupElement ] );
DeclareOperation( "IntoLoop", [ IsMagma ] );
DeclareOperation( "IntoGroup", [ IsMagma ] );
#############################################################################
## PRODUCTS OF QUASIGROUPS AND LOOPS
## --------------------------------------------------------------------------
DeclareGlobalFunction("ProductTableOfCanonicalCayleyTables");
#DirectProduct already declared for groups.
#############################################################################
## OPPOSITE QUASIGROUPS AND LOOPS
## --------------------------------------------------------------------------
DeclareGlobalFunction( "OppositeQuasigroup");
DeclareGlobalFunction( "OppositeLoop");
DeclareAttribute( "Opposite", IsMagma );
#############################################################################
## AUXILIARY
## --------------------------------------------------------------------------
DeclareGlobalFunction( "LOOPS_ReadCayleyTableFromFile" );
DeclareGlobalFunction( "LOOPS_CayleyTableByRightFolder" );
#############################################################################
##
#W quasigroups.gd Representing, creating and displaying quasigroups [loops]
##
#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 P. Vojtechovsky (University of Denver, USA)
##
#############################################################################
## GAP CATEGORIES AND REPRESENTATIONS
## -------------------------------------------------------------------------
## Categories convenient for defining quasigroups
## element which is an admissible argument for the right argument of /
DeclareCategory( "IsRightQuotientElement", IsExtLElement);
DeclareCategoryCollections("IsRightQuotientElement");
DeclareCategoryCollections("IsRightQuotientElementCollection");
## Every element with an inverse can form right quotients
## (in fact, in some sense it might be enough to have just a left inverse,
## but there doesn't seem to be any benefit to delving to that level of
## detail at this point.)
InstallTrueMethod(IsRightQuotientElement, IsMultiplicativeElementWithInverse);
## Now what we would like to do is re-declare
## DeclareOperation( "/", [IsExtRElement, IsRightQuotientElement] );
## but we can't since "/" is in the kernel, so we will have to content
## ourselves with InstallOtherMethod() calls on /. (I am not actually sure what
## the practical upshot of that is, i.e. if it has any shortcomings as compared
## to if we could declare "/" more generally.)
## Element which is admissible for the left argument of LeftQuotient()
DeclareCategory( "IsLeftQuotientElement", IsExtRElement);
DeclareCategoryCollections("IsLeftQuotientElement");
DeclareCategoryCollections("IsLeftQuotientElementCollection");
## Every element with an inverse can form left quotients
InstallTrueMethod(IsLeftQuotientElement, IsMultiplicativeElementWithInverse);
## Again, ideally (in some sense) we'd like to redeclare
## DeclareOperation("LeftQuotient", [IsLeftQuotientElement,IsExtLElement]);
## element of a quasigroup
DeclareSynonym( "IsQuasigroupElement",
IsMultiplicativeElement and
IsLeftQuotientElement and IsRightQuotientElement );
DeclareRepresentation( "IsQuasigroupElmRep",
IsPositionalObjectRep and IsMultiplicativeElement, [1] );
## element of a loop
DeclareSynonym( "IsLoopElement",
IsQuasigroupElement and IsMultiplicativeElementWithInverse );
DeclareRepresentation( "IsLoopElmRep",
IsPositionalObjectRep and IsMultiplicativeElementWithInverse, [1] );
## latin (auxiliary category for GAP to tell apart IsMagma and IsQuasigroup)
DeclareCategory( "IsLatinMagma", IsObject );
## quasigroup
DeclareCategory( "IsQuasigroup", IsMagma and IsLatinMagma );
## loop
DeclareSynonym( "IsLoop", IsQuasigroup and IsMagmaWithOne and
IsMultiplicativeElementWithInverseCollection);
#############################################################################
## TESTING MULTIPLICATION TABLES
## -------------------------------------------------------------------------
DeclareProperty( "IsLeftQuasigroupTable", IsMatrix );
DeclareProperty( "IsRightQuasigroupTable", IsMatrix );
DeclareSynonym( "IsQuasigroupTable",
IsLeftQuasigroupTable and IsRightQuasigroupTable );
DeclareSynonym( "IsQuasigroupCayleyTable", IsQuasigroupTable );
DeclareProperty( "IsLoopTable", IsMatrix );
DeclareSynonym( "IsLoopCayleyTable", IsLoopTable );
DeclareGlobalFunction("CanonicalCayleyTableOfLeftQuasigroupTable");
DeclareOperation( "CanonicalCayleyTable", [ IsMatrix ] );
DeclareOperation( "NormalizedQuasigroupTable", [ IsMatrix ] );
#############################################################################
## CREATING QUASIGROUPS AND LOOPS MANUALLY
## -------------------------------------------------------------------------
DeclareAttribute( "CayleyTable", IsMagma );
DeclareOperation( "QuasigroupByCayleyTable", [ IsMatrix ] );
DeclareOperation( "LoopByCayleyTable", [ IsMatrix ] );
DeclareOperation( "SpecifyElmNamePrefix", [ IsCollection, IsString ] );
DeclareSynonym( "SetQuasigroupElmName", SpecifyElmNamePrefix );
DeclareSynonym( "SetLoopElmName", SpecifyElmNamePrefix );
DeclareOperation( "BindElmNames", [ IsMagma ] );
DeclareAttribute( "ConstructorFromTable", IsMagma );
DeclareOperation( "CanonicalCopy", [ IsMagma ] );
#############################################################################
## CREATING QUASIGROUPS AND LOOPS FROM A FILE
## -------------------------------------------------------------------------
DeclareOperation( "QuasigroupFromFile", [ IsString, IsString ] );
DeclareOperation( "LoopFromFile", [ IsString, IsString ] );
#############################################################################
## CREATING QUASIGROUPS AND LOOPS BY SECTIONS
## -------------------------------------------------------------------------
DeclareGlobalFunction("CayleyTableByPerms");
DeclareOperation( "QuasigroupByLeftSection", [ IsPermCollection ] );
DeclareOperation( "LoopByLeftSection", [ IsPermCollection ] );
DeclareOperation( "QuasigroupByRightSection", [ IsPermCollection ] );
DeclareOperation( "LoopByRightSection", [ IsPermCollection ] );
DeclareOperation( "QuasigroupByRightFolder", [ IsGroup, IsGroup, IsMultiplicativeElementCollection ] );
DeclareOperation( "LoopByRightFolder", [ IsGroup, IsGroup, IsMultiplicativeElementCollection ] );
#############################################################################
## CONVERSIONS
## -------------------------------------------------------------------------
DeclareOperation( "IntoQuasigroup", [ IsMagma ] );
DeclareOperation( "PrincipalLoopIsotope",
[ IsQuasigroup, IsQuasigroupElement, IsQuasigroupElement ] );
DeclareOperation( "IntoLoop", [ IsMagma ] );
DeclareOperation( "IntoGroup", [ IsMagma ] );
#############################################################################
## PRODUCTS OF QUASIGROUPS AND LOOPS
## --------------------------------------------------------------------------
DeclareGlobalFunction("ProductTableOfCanonicalCayleyTables");
#DirectProduct already declared for groups.
#############################################################################
## OPPOSITE QUASIGROUPS AND LOOPS
## --------------------------------------------------------------------------
DeclareGlobalFunction( "OppositeQuasigroup");
DeclareGlobalFunction( "OppositeLoop");
DeclareAttribute( "Opposite", IsMagma );
#############################################################################
## AUXILIARY
## --------------------------------------------------------------------------
DeclareGlobalFunction( "LOOPS_ReadCayleyTableFromFile" );
DeclareGlobalFunction( "LOOPS_CayleyTableByRightFolder" );

48
gap/quasigroups.gi
View File

@ -893,32 +893,34 @@ end );
InstallMethod( ViewObj, "for loop",
[ IsLoop ],
function( L )
if HasIsAssociative( L ) and IsAssociative( L ) then
Print( "<associative loop of order ", Size( L ), ">");
elif HasIsExtraLoop( L ) and IsExtraLoop( L ) then
Print( "<extra loop of order ", Size( L ), ">");
elif HasIsMoufangLoop( L ) and IsMoufangLoop( L ) then
Print( "<Moufang loop of order ", Size( L ), ">");
elif HasIsCLoop( L ) and IsCLoop( L ) then
Print( "<C loop of order ", Size( L ), ">");
elif HasIsLeftBolLoop( L ) and IsLeftBolLoop( L ) then
Print( "<left Bol loop of order ", Size( L ), ">");
elif HasIsRightBolLoop( L ) and IsRightBolLoop( L ) then
Print( "<right Bol loop of order ", Size( L ), ">");
elif HasIsLCLoop( L ) and IsLCLoop( L ) then
Print( "<LC loop of order ", Size( L ), ">");
elif HasIsRCLoop( L ) and IsRCLoop( L ) then
Print( "<RC loop of order ", Size( L ), ">");
local PrintMe;
PrintMe := function( name, L )
Print( "<", name, " loop of order ", Size( L ), ">");
end;
if HasIsAssociative( L ) and IsAssociative( L ) then PrintMe( "associative", L );
elif HasIsExtraLoop( L ) and IsExtraLoop( L ) then PrintMe( "extra", L );
elif HasIsMoufangLoop( L ) and IsMoufangLoop( L ) then PrintMe( "Moufang", L );
elif HasIsCLoop( L ) and IsCLoop( L ) then PrintMe( "C", L );
elif HasIsLeftBruckLoop( L ) and IsLeftBruckLoop( L ) then PrintMe( "left Bruck", L );
elif HasIsRightBruckLoop( L ) and IsRightBruckLoop( L ) then PrintMe( "right Bruck", L );
elif HasIsLeftBolLoop( L ) and IsLeftBolLoop( L ) then PrintMe( "left Bol", L );
elif HasIsRightBolLoop( L ) and IsRightBolLoop( L ) then PrintMe( "right Bol", L );
elif HasIsAutomorphicLoop( L ) and IsAutomorphicLoop( L ) then PrintMe( "automorphic", 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
if HasIsRightAlternative( L ) and IsRightAlternative( L ) then
Print( "<alternative loop of order ", Size( L ), ">");
else
Print( "<left alternative loop of order ", Size( L ), ">");
PrintMe("alternative", L );
else
PrintMe("left alternative", L );
fi;
elif HasIsRightAlternative( L ) and IsRightAlternative( L ) then
Print( "<right alternative loop of order ", Size( L ), ">");
elif HasIsFlexible( L ) and IsFlexible( L ) then
Print( "<flexible loop of order ", Size( L ), ">");
elif HasIsRightAlternative( L ) and IsRightAlternative( L ) then PrintMe( "right alternative", L );
elif HasIsCommutative( L ) and IsCommutative( L ) then PrintMe( "commutative", L );
elif HasIsFlexible( L ) and IsFlexible( L ) then PrintMe( "flexible", L);
else
# MORE ??
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> AssociatedRightBruckLoop( Q );
<right Bol loop of order 15>
<right Bruck loop of order 15>
# 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
##
#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 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
##
#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 P. Vojtechovsky (University of Denver, USA)
@ -34,6 +34,11 @@ Group([ (1,2,3), (1,3,2) ])
gap> Q := DirectProduct( MoufangLoop( 32, 5 ) );;
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 );
(2,3,4)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)
gap> LoopsUpToIsomorphism( [Q,Qp] );

54
tst/lib.tst
View File

@ -2,7 +2,7 @@
##
#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 P. Vojtechovsky (University of Denver, USA)
@ -156,19 +156,34 @@ gap> LCCLoop(6,3); LCCLoop(25,119);
# CC LOOPS
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.
------
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
28 loops of order 16
42 loops of order 16
7 loops of order 18
3 loops of order 20
1 loop of order 21
14 loops of order 24
55 loops of order 27
3 loops of order p^2 for every odd prime p,
5 loops of order 25
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
true
@ -233,10 +248,7 @@ gap> CodeLoop( 64, 80 );
gap> DisplayLibraryInfo("automorphic");
The library contains:
- 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 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.
- all commutative automorphic loops of order 3, 9, 27, 81.
------
Extent of the library:
1 loop of order 3
@ -249,7 +261,6 @@ Extent of the library:
2 loops of order 15
7 loops of order 27
72 loops of order 81
118451 loops of order 243
true
gap> AutomorphicLoop(15,2);
@ -258,7 +269,24 @@ gap> AutomorphicLoop(15,2);
gap> AutomorphicLoop(27,1);
<automorphic loop 27/1>
gap> AutomorphicLoop(243,100);
<automorphic loop 243/100>
gap> AutomorphicLoop(81,10);
<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 );

12
tst/testall.g
View File

@ -2,15 +2,15 @@
##
#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 P. Vojtechovsky (University of Denver, USA)
##
dirs := DirectoriesPackageLibrary( "loops", "tst" );
ReadTest( Filename( dirs, "core_methods.tst" ) );
ReadTest( Filename( dirs, "nilpot.tst" ) );
ReadTest( Filename( dirs, "iso.tst" ) );
ReadTest( Filename( dirs, "lib.tst" ) );
ReadTest( Filename( dirs, "bol.tst" ) );
Test( Filename( dirs, "core_methods.tst" ), rec( compareFunction := "uptowhitespace" ) );
Test( Filename( dirs, "nilpot.tst" ), rec( compareFunction := "uptowhitespace" ) );
Test( Filename( dirs, "iso.tst" ), rec( compareFunction := "uptowhitespace" ) );
Test( Filename( dirs, "lib.tst" ), rec( compareFunction := "uptowhitespace" ) );
Test( Filename( dirs, "bol.tst" ), rec( compareFunction := "uptowhitespace" ) );