loops/doc/chap6.txt

554 lines
30 KiB
Plaintext
Raw Normal View History

2017-10-16 19:43:09 +00:00
6 Methods Based on Permutation Groups
Most calculations in the LOOPS package are delegated to groups, taking
advantage of the various permutations and permutation groups associated with
quasigroups. This chapter explains in detail how the permutations associated
with a quasigroup are calculated, and it also describes some of the core
methods of LOOPS based on permutations. Additional core methods can be found
in Chapter 7.
6.1 Parent of a Quasigroup
Let Q be a quasigroup and S a subquasigroup of Q. Since the multiplication
in S coincides with the multiplication in Q, it is reasonable not to store
the multiplication table of S. However, the quasigroup S then must know that
it is a subquasigroup of Q.
6.1-1 Parent
Parent( Q )  attribute
Returns: The parent quasigroup of the quasigroup Q.
When Q is not created as a subquasigroup of another quasigroup, the
attribute Parent(Q) is set to Q. When Q is created as a subquasigroup of a
quasigroup H, we set Parent(Q) equal to Parent(H). Thus, in effect,
Parent(Q) is the largest quasigroup from which Q has been created.
6.1-2 Position
Position( Q, x )  operation
Returns: The position of x among the elements of Q.
Let Q be a quasigroup with parent P, where P is some n-element quasigroup.
Let x be an element of Q. Then x![1] is the position of x among the elements
of P, i.e., x![1] = Position(Elements(P),x).
While referring to elements of Q by their positions, the user should
understand whether the positions are meant among the elements of Q, or among
the elements of the parent P of Q. Since it requires no calculation to
obtain x![1], we always use the position of an element in its parent
quasigroup in LOOPS. In this way, many attributes of a quasigroup, including
its Cayley table, are permanently tied to its parent.
It is now clear why we have not insisted that Cayley tables of quasigroups
must have entries covering the entire interval 1, dots, n for some n.
6.1-3 PosInParent
PosInParent( S )  operation
Returns: When S is a list of quasigroup elements (not necessarily from the
same quasigroup), returns the corresponding list of positions of
elements of S in the corresponding parent, i.e., PosInParent(S)[i]
= S[i]![1] = Position(Parent(S[i]),S[i]).
Quasigroups with the same parent can be compared as follows. Assume that A,
B are two quasigroups with common parent Q. Let G_A, G_B be the canonical
generating sets of A and B, respectively, obtained by the method
GeneratorsSmallest (see Section 5.5). Then we define A<B if and only if
G_A<G_B lexicographically.
6.2 Subquasigroups and Subloops
6.2-1 Subquasigroup
Subquasigroup( Q, S )  operation
Returns: When S is a subset of elements or indices of a quasigroup (resp.
loop) Q, returns the smallest subquasigroup (resp. subloop) of Q
containing S.
We allow S to be a list of elements of Q, or a list of integers representing
the positions of the respective elements in the parent quasigroup (resp.
loop) of Q.
If S is empty, Subquasigroup(Q,S) returns the empty set if Q is a
quasigroup, and it returns the one-element subloop of Q if Q is a loop.
Remark: The empty set is sometimes considered to be a subquasigroup of Q
(although not in LOOPS). The above convention is useful for handling certain
situations, for instance when the user calls Center(Q) for a quasigroup Q
with empty center.
6.2-2 Subloop
Subloop( Q, S )  operation
This is an analog of Subquasigroup(Q,S) that can be used only when Q is a
loop. Since there is no difference in the outcome while calling
Subquasigroup(Q,S) or Subloop(Q,S) when Q is a loop, it is safe to always
call Subquasigroup(Q,S), whether Q is a loop or not.
6.2-3 IsSubquasigroup and IsSubloop
IsSubquasigroup( Q, S )  operation
IsSubloop( Q, S )  operation
Returns: true if S is a subquasigroup (resp. subloop) of a quasigroup
(resp. loop) Q, false otherwise. In other words, returns true if S
and Q are quasigroups (resp. loops) with the same parent and S is
a subset of Q.
6.2-4 AllSubquasigroups
AllSubquasigroups( Q )  operation
Returns: A list of all subquasigroups of a loop Q.
6.2-5 AllSubloops
AllSubloops( Q )  operation
Returns: A list of all subloops of a loop Q.
6.2-6 RightCosets
RightCosets( Q, S )  function
Returns: If S is a subloop of Q, returns a list of all right cosets of S in
Q.
The coset S is listed first, and the elements of each coset are ordered in
the same way as the elements of S, i.e., if S= [s_1,dots,s_m], then
Sx=[s_1x,dots,s_mx].
6.2-7 RightTransversal
RightTransversal( Q, S )  operation
Returns: A right transversal of a subloop S in a loop Q. The transversal
consists of the list of first elements from the right cosets
obtained by RightCosets(Q,S).
When S is a subloop of Q, the right transversal of S with respect to Q is a
subset of Q containing one element from each right coset of S in Q.
6.3 Translations and Sections
When x is an element of a quasigroup Q, the left translation L_x is a
permutation of Q. In LOOPS, all permutations associated with quasigroups and
their elements are permutations in the sense of GAP, i.e., they are
bijections of some interval 1, dots, n. Moreover, following our convention,
the numerical entries of the permutations point to the positions among
elements of the parent of Q, not among elements of Q.
6.3-1 LeftTranslation and RightTranslation
LeftTranslation( Q, x )  operation
RightTranslation( Q, x )  operation
Returns: If x is an element of a quasigroup Q, returns the left translation
(resp. right translation) by x in Q.
6.3-2 LeftSection and RightSection
LeftSection( Q )  operation
RightSection( Q )  operation
Returns: The left section (resp. right section) of a quasigroup Q.
Here is an example illustrating the main features of the subquasigroup
construction and the relationship between a quasigroup and its parent.
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.
 Example 
gap> M := MoufangLoop( 12, 1 );; S := Subloop( M, [ M.5 ] ); 
<loop of order 3> 
gap> [ Parent( S ) = M, Elements( S ), PosInParent( S ) ]; 
[ true, [ l1, l3, l5], [ 1, 3, 5 ] ] 
gap> HasCayleyTable( S ); 
false 
gap> SetLoopElmName( S, "s" );; Elements( S ); Elements( M ); 
[ s1, s3, s5 ] 
[ s1, s2, s3, s4, s5, s6, s7, s8, s9, s10, s11, s12 ] 
gap> CayleyTable( S ); 
[ [ 1, 3, 5 ], [ 3, 5, 1 ], [ 5, 1, 3 ] ] 
gap> LeftSection( S ); 
[ (), (1,3,5), (1,5,3) ] 
gap> [ HasCayleyTable( S ), Parent( S ) = M ]; 
[ true, true ] 
gap> L := LoopByCayleyTable( CayleyTable( S ) );; Elements( L ); 
[ l1, l2, l3 ] 
gap> [ Parent( L ) = L, IsSubloop( M, S ), IsSubloop( M, L ) ]; 
[ true, true, false ] 
gap> LeftSection( L ); 
[ (), (1,2,3), (1,3,2) ] 

6.4 Multiplication Groups
6.4-1 LeftMutliplicationGroup, RightMultiplicationGroup and
MultiplicationGroup
LeftMultiplicationGroup( Q )  attribute
RightMultiplicationGroup( Q )  attribute
MultiplicationGroup( Q )  attribute
Returns: The left multiplication group, right multiplication group, resp.
multiplication group of a quasigroup Q.
6.4-2 RelativeLeftMultiplicationGroup, RelativeRightMultiplicationGroup and
RelativeMultiplicationGroup
RelativeLeftMultiplicationGroup( Q, S )  operation
RelativeRightMultiplicationGroup( Q, S )  operation
RelativeMultiplicationGroup( Q, S )  operation
Returns: The relative left multiplication group, the relative right
multiplication group, resp. the relative multiplication group of a
quasigroup Q with respect to a subquasigroup S of Q.
Let S be a subquasigroup of a quasigroup Q. Then the relative left
multiplication group of Q with respect to S is the group ⟨ L(x)|x∈ S⟩, where
L(x) is the left translation by x in Q restricted to S. The relative right
multiplication group and the relative multiplication group are defined
analogously.
6.5 Inner Mapping Groups
By a result of Bruck, the left inner mapping group of a loop is generated by
all left inner mappings L(x,y) = L_yx^-1L_yL_x, and the right inner mapping
group is generated by all right inner mappings R(x,y) = R_xy^-1R_yR_x.
In analogy with group theory, we define the conjugations or the middle inner
mappings as T(x) = L_x^-1R_x. The middle inner mapping grroup is then the
group generated by all conjugations.
6.5-1 LeftInnerMapping, RightInnerMapping, MiddleInnerMapping
LeftInnerMapping( Q, x, y )  operation
RightInnerMapping( Q, x, y )  operation
MiddleInnerMapping( Q, x )  operation
Returns: The left inner mapping L(x,y), the right inner mapping R(x,y),
resp. the middle inner mapping T(x) of a loop Q.
6.5-2 LeftInnerMappingGroup, RightInnerMappingGroup, MiddleInnerMappingGroup
LeftInnerMappingGroup( Q )  attribute
RightInnerMappingGroup( Q )  attribute
MiddleInnerMappingGroup( Q )  attribute
Returns: The left inner mapping group, right inner mapping group, resp.
middle inner mapping group of a loop Q.
6.5-3 InnerMappingGroup
InnerMappingGroup( Q )  attribute
Returns: The inner mapping group of a loop Q.
Here is an example for multiplication groups and inner mapping groups:
 Example 
gap> M := MoufangLoop(12,1); 
<Moufang loop 12/1> 
gap> LeftSection(M)[2]; 
(1,2)(3,4)(5,6)(7,8)(9,12)(10,11) 
gap> Mlt := MultiplicationGroup(M); Inn := InnerMappingGroup(M); 
<permutation group of size 2592 with 23 generators> 
Group([ (4,6)(7,11), (7,11)(8,10), (2,6,4)(7,9,11), (3,5)(9,11), (8,12,10) ]) 
gap> Size(Inn); 
216 

6.6 Nuclei, Commutant, Center, and Associator Subloop
See Section 2.3 for the relevant definitions.
6.6-1 LeftNucles, MiddleNucleus, and RightNucleus
LeftNucleus( Q )  attribute
MiddleNucleus( Q )  attribute
RightNucleus( Q )  attribute
Returns: The left nucleus, middle nucleus, resp. right nucleus of a
quasigroup Q.
6.6-2 Nuc, NucleusOfQuasigroup and NucleusOfLoop
Nuc( Q )  attribute
NucleusOfQuasigroup( Q )  attribute
NucleusOfLoop( Q )  attribute
Returns: These synonymous attributes return the nucleus of a quasigroup Q.
Since all nuclei are subquasigroups of Q, they are returned as
subquasigroups (resp. subloops). When Q is a loop then all nuclei are in
fact groups, and they are returned as associative loops.
Remark: The name Nucleus is a global function of GAP with two variables. We
have therefore used Nuc rather than Nucleus for the nucleus. This
abbreviation is sometimes used in the literature, too.
6.6-3 Commutant
Commutant( Q )  attribute
Returns: The commutant of a quasigroup Q.
6.6-4 Center
Center( Q )  attribute
Returns: The center of a quasigroup Q.
If Q is a loop, the center of Q is a subgroup of Q and it is returned as an
associative loop.
6.6-5 AssociatorSubloop
AssociatorSubloop( Q )  attribute
Returns: The associator subloop of a loop Q.
We calculate the associator subloop of Q as the smallest normal subloop of Q
containing all elements xbackslashα(x), where x is an element of Q and α is
a left inner mapping of Q.
6.7 Normal Subloops and Simple Loops
6.7-1 IsNormal
IsNormal( Q, S )  operation
Returns: true if S is a normal subloop of a loop Q.
A subloop S of a loop Q is normal if it is invariant under all inner
mappings of Q.
6.7-2 NormalClosure
NormalClosure( Q, S )  operation
Returns: The normal closure of a subset S of a loop Q.
For a subset S of a loop Q, the normal closure of S in Q is the smallest
normal subloop of Q containing S.
6.7-3 IsSimple
IsSimple( Q )  operation
Returns: true if Q is a simple loop.
A loop Q is simple if {1} and Q are the only normal subloops of Q.
6.8 Factor Loops
6.8-1 FactorLoop
FactorLoop( Q, S )  operation
Returns: When S is a normal subloop of a loop Q, returns the factor loop
Q/S.
6.8-2 NaturalHomomorphismByNormalSubloop
NaturalHomomorphismByNormalSubloop( Q, S )  operation
Returns: When S is a normal subloop of a loop Q, returns the natural
projection from Q onto Q/S.
 Example 
gap> M := MoufangLoop( 12, 1 );; S := Subloop( M, [ M.3 ] ); 
<loop of order 3> 
gap> IsNormal( M, S ); 
true 
gap> F := FactorLoop( M, S ); 
<loop of order 4> 
gap> NaturalHomomorphismByNormalSubloop( M, S ); 
MappingByFunction( <loop of order 12>, <loop of order 4>, 
 function( x ) ... end ) 

6.9 Nilpotency and Central Series
See Section 2.4 for the relevant definitions.
6.9-1 IsNilpotent
IsNilpotent( Q )  property
Returns: true if Q is a nilpotent loop.
6.9-2 NilpotencyClassOfLoop
NilpotencyClassOfLoop( Q )  attribute
Returns: The nilpotency class of a loop Q if Q is nilpotent, fail
otherwise.
6.9-3 IsStronglyNilpotent
IsStronglyNilpotent( Q )  property
Returns: true if Q is a strongly nilpotent loop.
A loop Q is said to be strongly nilpotent if its multiplication group is
nilpotent.
6.9-4 UpperCentralSeries
UpperCentralSeries( Q )  attribute
Returns: When Q is a nilpotent loop, returns the upper central series of Q,
else returns fail.
6.9-5 LowerCentralSeries
LowerCentralSeries( Q )  attribute
Returns: When Q is a nilpotent loop, returns the lower central series of Q,
else returns fail.
The lower central series for loops is defined analogously to groups.
6.10 Solvability, Derived Series and Frattini Subloop
See Section 2.4 for definitions of solvability an derived subloop.
6.10-1 IsSolvable
IsSolvable( Q )  property
Returns: true if Q is a solvable loop.
6.10-2 DerivedSubloop
DerivedSubloop( Q )  attribute
Returns: The derived subloop of a loop Q.
6.10-3 DerivedLength
DerivedLength( Q )  attribute
Returns: If Q is solvable, returns the derived length of Q, else returns
fail.
6.10-4 FrattiniSubloop and FrattinifactorSize
FrattiniSubloop( Q )  attribute
Returns: The Frattini subloop of Q. The method is implemented only for
strongly nilpotent loops.
Frattini subloop of a loop Q is the intersection of maximal subloops of Q.
6.10-5 FrattinifactorSize
FrattinifactorSize( Q )  attribute
6.11 Isomorphisms and Automorphisms
6.11-1 IsomorphismQuasigroups
IsomorphismQuasigroups( Q, L )  operation
Returns: An isomorphism from a quasigroup Q to a quasigroup L if the
quasigroups are isomorphic, fail otherwise.
If an isomorphism exists, it is returned as a permutation f of 1,dots,|Q|,
where i^f=j means that the ith element of Q is mapped onto the jth element
of L. Note that this convention is used even if the underlying sets of Q, L
are not indexed by consecutive integers.
6.11-2 IsomorphismLoops
IsomorphismLoops( Q, L )  operation
Returns: An isomorphism from a loop Q to a loop L if the loops are
isomorphic, fail otherwise, with the same convention as in
IsomorphismQuasigroups.
6.11-3 QuasigroupsUpToIsomorphism
QuasigroupsUpToIsomorphism( ls )  operation
Returns: Given a list ls of quasigroups, returns a sublist of ls consisting
of representatives of isomorphism classes of quasigroups from ls.
6.11-4 LoopsUpToIsomorphism
LoopsUpToIsomorphism( ls )  operation
Returns: Given a list ls of loops, returns a sublist of ls consisting of
representatives of isomorphism classes of loops from ls.
6.11-5 AutomorphismGroup
AutomorphismGroup( Q )  attribute
Returns: The automorphism group of a loop or quasigroups Q, with the same
convention on permutations as in IsomorphismQuasigroups.
Remark: Since two isomorphisms differ by an automorphism, all isomorphisms
from Q to L can be obtained by a combination of IsomorphismLoops(Q,L) (or
IsomorphismQuasigroups(Q,L)) and AutomorphismGroup(L).
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. LOOPS contains several functions for this purpose.
6.11-6 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.
6.11-7 IsomorphicCopyByNormalSubloop
IsomorphicCopyByNormalSubloop( Q, S )  operation
Returns: When S is a normal subloop of a loop Q, returns an isomorphic copy
of Q in which the elements are ordered according to the right
cosets of S. In particular, the Cayley table of S will appear in
the top left corner of the Cayley table of the resulting loop.
In order to speed up the search for isomorphisms and automorphisms, we first
calculate some loop invariants preserved under isomorphisms, and then we use
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
Discriminator( Q )  operation
Returns: A data structure with isomorphism invariants of a loop Q.
See [Voj06] or the file iso.gi for more details. The format of the
discriminator has been changed from version 3.2.0 up to accommodate
isomorphism searches for quasigroups.
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
AreEqualDiscriminators( D1, D2 )  operation
Returns: true if D1, D2 are equal discriminators for the purposes of
isomorphism searches.
6.12 Isotopisms
At the moment, LOOPS contains only slow methods for testing if two loops are
isotopic. The method works as follows: It is well known that if a loop K is
isotopic to a loop L then there exist a principal loop isotope P of K such
that P is isomorphic to L. The algorithm first finds all principal isotopes
of K, then filters them up to isomorphism, and then checks if any of them is
isomorphic to L. This is rather slow already for small orders.
6.12-1 IsotopismLoops
IsotopismLoops( K, L )  operation
Returns: fail if K, L are not isotopic loops, else it returns an isotopism
as a triple of bijections on 1,dots,|K|.
6.12-2 LoopsUpToIsotopism
LoopsUpToIsotopism( ls )  operation
Returns: Given a list ls of loops, returns a sublist of ls consisting of
representatives of isotopism classes of loops from ls.