loops/doc/chap6.txt

  
  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 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:  LoopIsomorphism(Q,f)      if      Q     is     a     loop,     and
            QuasigroupIsomorphism(Q,f) if Q is a quasigroup.
  
  6.11-9 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.
  
  6.11-10 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.
  
  6.11-11 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.