  
  [1X6 [33X[0;0YMethods Based on Permutation Groups[133X[101X
  
  [33X[0;0YMost  calculations  in  the  [5XLOOPS[105X  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 [5XLOOPS[105X based on permutations. Additional core methods can be found
  in Chapter [14X7[114X.[133X
  
  
  [1X6.1 [33X[0;0YParent of a Quasigroup[133X[101X
  
  [33X[0;0YLet  [22XQ[122X  be a quasigroup and [22XS[122X a subquasigroup of [22XQ[122X. Since the multiplication
  in  [22XS[122X  coincides with the multiplication in [22XQ[122X, it is reasonable not to store
  the multiplication table of [22XS[122X. However, the quasigroup [22XS[122X then must know that
  it is a subquasigroup of [22XQ[122X.[133X
  
  [1X6.1-1 Parent[101X
  
  [29X[2XParent[102X( [3XQ[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10YThe parent quasigroup of the quasigroup [3XQ[103X.[133X
  
  [33X[0;0YWhen  [3XQ[103X  is  not  created  as  a  subquasigroup  of  another quasigroup, the
  attribute  [10XParent([3XQ[103X[10X)[110X  is set to [3XQ[103X. When [3XQ[103X is created as a subquasigroup of a
  quasigroup  [3XH[103X,  we  set  [10XParent([3XQ[103X[10X)[110X  equal  to  [10XParent([3XH[103X[10X)[110X.  Thus,  in effect,
  [10XParent([3XQ[103X[10X)[110X is the largest quasigroup from which [3XQ[103X has been created.[133X
  
  [1X6.1-2 Position[101X
  
  [29X[2XPosition[102X( [3XQ[103X, [3Xx[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10YThe position of [3Xx[103X among the elements of [3XQ[103X.[133X
  
  [33X[0;0YLet  [3XQ[103X  be a quasigroup with parent [3XP[103X, where [3XP[103X is some [22Xn[122X-element quasigroup.
  Let [3Xx[103X be an element of [3XQ[103X. Then [10X[3Xx[103X[10X![1][110X is the position of [3Xx[103X among the elements
  of [3XP[103X, i.e., [10X[3Xx[103X[10X![1] = Position(Elements([3XP[103X[10X),[3Xx[103X[10X)[110X.[133X
  
  [33X[0;0YWhile  referring  to  elements  of  [3XQ[103X  by  their  positions, the user should
  understand whether the positions are meant among the elements of [3XQ[103X, or among
  the  elements  of  the  parent  [3XP[103X  of [3XQ[103X. Since it requires no calculation to
  obtain  [10X[3Xx[103X[10X![1][110X,  we  always  use  the  position  of  an element in its parent
  quasigroup in [5XLOOPS[105X. In this way, many attributes of a quasigroup, including
  its Cayley table, are permanently tied to its parent.[133X
  
  [33X[0;0YIt  is  now clear why we have not insisted that Cayley tables of quasigroups
  must have entries covering the entire interval [22X1[122X, [22Xdots[122X, [22Xn[122X for some [22Xn[122X.[133X
  
  [1X6.1-3 PosInParent[101X
  
  [29X[2XPosInParent[102X( [3XS[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10YWhen  [3XS[103X is a list of quasigroup elements (not necessarily from the
            same  quasigroup),  returns the corresponding list of positions of
            elements of [3XS[103X in the corresponding parent, i.e., [10XPosInParent([3XS[103X[10X)[i]
            = [3XS[103X[10X[i]![1] = Position(Parent([3XS[103X[10X[i]),[3XS[103X[10X[i])[110X.[133X
  
  [33X[0;0YQuasigroups  with the same parent can be compared as follows. Assume that [22XA[122X,
  [22XB[122X  are  two  quasigroups with common parent [22XQ[122X. Let [22XG_A[122X, [22XG_B[122X be the canonical
  generating   sets   of  [22XA[122X  and  [22XB[122X,  respectively,  obtained  by  the  method
  [10XGeneratorsSmallest[110X  (see  Section  [14X5.5[114X).  Then  we define [22XA<B[122X if and only if
  [22XG_A<G_B[122X lexicographically.[133X
  
  
  [1X6.2 [33X[0;0YSubquasigroups and Subloops[133X[101X
  
  [1X6.2-1 Subquasigroup[101X
  
  [29X[2XSubquasigroup[102X( [3XQ[103X, [3XS[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10YWhen  [3XS[103X  is  a  subset (of elements or indices) of a quasigroup [3XQ[103X,
            returns the smallest subquasigroup (resp. subloop) of [3XQ[103X containing
            [3XS[103X.[133X
  
  [33X[0;0YWe allow [3XS[103X to be a list of elements of [3XQ[103X, or a list of integers representing
  the  positions  of  the  respective elements in the parent quasigroup (resp.
  loop) of [3XQ[103X.[133X
  
  [33X[0;0YIf  [3XS[103X  is  empty,  [10XSubloop([3XQ[103X[10X,[3XS[103X[10X)[110X  returns the one-element subloop of [3XQ[103X, while
  [10XSubquasigroup([3XQ[103X[10X,[3XS[103X[10X)[110X returns the empty set.[133X
  
  [33X[0;0Y[12XRemark:[112X  The  empty  set  is sometimes considered to be a subquasigroup of [3XQ[103X
  (although not in [5XLOOPS[105X). The above convention is useful for handling certain
  situations,  for  instance  when the user calls [10XCenter([3XQ[103X[10X)[110X for a quasigroup [3XQ[103X
  with empty center.[133X
  
  [1X6.2-2 Subloop[101X
  
  [29X[2XSubloop[102X( [3XQ[103X, [3XS[103X ) [32X operation
  
  [33X[0;0YThis is a loop analog of [10XIsSubquasigroup([3XQ[103X[10X,[3XS[103X[10X)[110X.[133X
  
  
  [1X6.2-3 [33X[0;0YIsSubquasigroup and IsSubloop[133X[101X
  
  [29X[2XIsSubquasigroup[102X( [3XQ[103X, [3XS[103X ) [32X operation
  [29X[2XIsSubloop[102X( [3XQ[103X, [3XS[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X  if  [3XS[103X  is  a  subquasigroup  (resp. subloop) of a quasigroup
            (resp. loop) [3XQ[103X, [10Xfalse[110X otherwise. In other words, returns [10Xtrue[110X if [3XS[103X
            and  [3XQ[103X are quasigroups (resp. loops) with the same parent and [3XS[103X is
            a subset of [3XQ[103X.[133X
  
  [1X6.2-4 AllSubloops[101X
  
  [29X[2XAllSubloops[102X( [3XQ[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10YA list of all subloops of a loop [3XQ[103X.[133X
  
  [1X6.2-5 RightCosets[101X
  
  [29X[2XRightCosets[102X( [3XQ[103X, [3XS[103X ) [32X function
  [6XReturns:[106X  [33X[0;10YIf [3XS[103X is a subloop of [3XQ[103X, returns a list of all right cosets of [3XS[103X in
            [3XQ[103X.[133X
  
  [33X[0;0YThe  coset  [3XS[103X is listed first, and the elements of each coset are ordered in
  the  same  way  as  the  elements  of  [3XS[103X,  i.e.,  if [3XS[103X[22X= [s_1,dots,s_m][122X, then
  [3XS[103X[22Xx=[s_1x,dots,s_mx][122X.[133X
  
  [1X6.2-6 RightTransversal[101X
  
  [29X[2XRightTransversal[102X( [3XQ[103X, [3XS[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10YA  right  transversal  of a subloop [3XS[103X in a loop [3XQ[103X. The transversal
            consists  of  the  list  of  first  elements from the right cosets
            obtained by [10XRightCosets([3XQ[103X[10X,[3XS[103X[10X)[110X.[133X
  
  [33X[0;0YWhen  [3XS[103X is a subloop of [3XQ[103X, the right transversal of [3XS[103X with respect to [3XQ[103X is a
  subset of [3XQ[103X containing one element from each right coset of [3XS[103X in [3XQ[103X.[133X
  
  
  [1X6.3 [33X[0;0YTranslations and Sections[133X[101X
  
  [33X[0;0YWhen  [22Xx[122X  is  an  element  of  a  quasigroup [22XQ[122X, the left translation [22XL_x[122X is a
  permutation of [22XQ[122X. In [5XLOOPS[105X, all permutations associated with quasigroups and
  their  elements  are  permutations  in  the  sense  of  [5XGAP[105X,  i.e., they are
  bijections  of some interval [22X1[122X, [22Xdots[122X, [22Xn[122X. Moreover, following our convention,
  the  numerical  entries  of  the  permutations  point to the positions among
  elements of the parent of [22XQ[122X, not among elements of [22XQ[122X.[133X
  
  
  [1X6.3-1 [33X[0;0YLeftTranslation and RightTranslation[133X[101X
  
  [29X[2XLeftTranslation[102X( [3XQ[103X, [3Xx[103X ) [32X operation
  [29X[2XRightTranslation[102X( [3XQ[103X, [3Xx[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10YIf [3Xx[103X is an element of a quasigroup [3XQ[103X, returns the left translation
            (resp. right translation) by [3Xx[103X in [3XQ[103X.[133X
  
  
  [1X6.3-2 [33X[0;0YLeftSection and RightSection[133X[101X
  
  [29X[2XLeftSection[102X( [3XQ[103X ) [32X operation
  [29X[2XRightSection[102X( [3XQ[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10YThe left section (resp. right section) of a quasigroup [3XQ[103X.[133X
  
  [33X[0;0YHere  is  an  example  illustrating  the  main features of the subquasigroup
  construction and the relationship between a quasigroup and its parent.[133X
  
  [33X[0;0YNote  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.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XM := MoufangLoop( 12, 1 );; S := Subloop( M, [ M.5 ] );
[127X[104X
    [4X[28X<loop of order 3>
[128X[104X
    [4X[25Xgap>[125X [27X[ Parent( S ) = M, Elements( S ), PosInParent( S ) ];
[127X[104X
    [4X[28X[ true, [ l1, l3, l5], [ 1, 3, 5 ] ]
[128X[104X
    [4X[25Xgap>[125X [27XHasCayleyTable( S );
[127X[104X
    [4X[28Xfalse
[128X[104X
    [4X[25Xgap>[125X [27XSetLoopElmName( S, "s" );; Elements( S ); Elements( M );
[127X[104X
    [4X[28X[ s1, s3, s5 ]
[128X[104X
    [4X[28X[ s1, s2, s3, s4, s5, s6, s7, s8, s9, s10, s11, s12 ]
[128X[104X
    [4X[25Xgap>[125X [27XCayleyTable( S );
[127X[104X
    [4X[28X[ [ 1, 3, 5 ], [ 3, 5, 1 ], [ 5, 1, 3 ] ]
[128X[104X
    [4X[25Xgap>[125X [27XLeftSection( S );
[127X[104X
    [4X[28X[ (), (1,3,5), (1,5,3) ]
[128X[104X
    [4X[25Xgap>[125X [27X[ HasCayleyTable( S ), Parent( S ) = M ];
[127X[104X
    [4X[28X[ true, true ]
[128X[104X
    [4X[25Xgap>[125X [27XL := LoopByCayleyTable( CayleyTable( S ) );; Elements( L );
[127X[104X
    [4X[28X[ l1, l2, l3 ]
[128X[104X
    [4X[25Xgap>[125X [27X[ Parent( L ) = L, IsSubloop( M, S ), IsSubloop( M, L ) ];
[127X[104X
    [4X[28X[ true, true, false ]
[128X[104X
    [4X[25Xgap>[125X [27XLeftSection( L );
[127X[104X
    [4X[28X[ (), (1,2,3), (1,3,2) ]
[128X[104X
  [4X[32X[104X
  
  
  [1X6.4 [33X[0;0YMultiplication Groups[133X[101X
  
  
  [1X6.4-1       [33X[0;0YLeftMutliplicationGroup,       RightMultiplicationGroup      and[101X
  [1XMultiplicationGroup[133X[101X
  
  [29X[2XLeftMultiplicationGroup[102X( [3XQ[103X ) [32X attribute
  [29X[2XRightMultiplicationGroup[102X( [3XQ[103X ) [32X attribute
  [29X[2XMultiplicationGroup[102X( [3XQ[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10YThe  left  multiplication group, right multiplication group, resp.
            multiplication group of a quasigroup [3XQ[103X.[133X
  
  
  [1X6.4-2  [33X[0;0YRelativeLeftMultiplicationGroup, RelativeRightMultiplicationGroup and[101X
  [1XRelativeMultiplicationGroup[133X[101X
  
  [29X[2XRelativeLeftMultiplicationGroup[102X( [3XQ[103X, [3XS[103X ) [32X operation
  [29X[2XRelativeRightMultiplicationGroup[102X( [3XQ[103X, [3XS[103X ) [32X operation
  [29X[2XRelativeMultiplicationGroup[102X( [3XQ[103X, [3XS[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10YThe   relative  left  multiplication  group,  the  relative  right
            multiplication group, resp. the relative multiplication group of a
            quasigroup [3XQ[103X with respect to a subquasigroup [3XS[103X of [3XQ[103X.[133X
  
  [33X[0;0YLet  [22XS[122X  be  a  subquasigroup  of  a  quasigroup  [22XQ[122X.  Then  the [13Xrelative left
  multiplication group[113X of [22XQ[122X with respect to [22XS[122X is the group [22X⟨ L(x)|x∈ S⟩[122X, where
  [22XL(x)[122X  is  the left translation by [22Xx[122X in [22XQ[122X restricted to [22XS[122X. The [13Xrelative right
  multiplication  group[113X  and  the  [13Xrelative  multiplication  group[113X are defined
  analogously.[133X
  
  
  [1X6.5 [33X[0;0YInner Mapping Groups[133X[101X
  
  [33X[0;0YBy a result of Bruck, the left inner mapping group of a loop is generated by
  all  [13Xleft inner mappings[113X [22XL(x,y) = L_yx^-1L_yL_x[122X, and the right inner mapping
  group is generated by all [13Xright inner mappings[113X [22XR(x,y) = R_xy^-1R_yR_x[122X.[133X
  
  [33X[0;0YIn analogy with group theory, we define the [13Xconjugations[113X or the [13Xmiddle inner
  mappings[113X  as  [22XT(x)  = L_x^-1R_x[122X. The [13Xmiddle inner mapping grroup[113X is then the
  group generated by all conjugations.[133X
  
  
  [1X6.5-1 [33X[0;0YLeftInnerMapping, RightInnerMapping, MiddleInnerMapping[133X[101X
  
  [29X[2XLeftInnerMapping[102X( [3XQ[103X, [3Xx[103X, [3Xy[103X ) [32X operation
  [29X[2XRightInnerMapping[102X( [3XQ[103X, [3Xx[103X, [3Xy[103X ) [32X operation
  [29X[2XMiddleInnerMapping[102X( [3XQ[103X, [3Xx[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10YThe  left  inner  mapping  [22XL([122X[3Xx[103X,[3Xy[103X[22X)[122X, the right inner mapping [22XR([122X[3Xx[103X,[3Xy[103X[22X)[122X,
            resp. the middle inner mapping [22XT([122X[3Xx[103X[22X)[122X of a loop [3XQ[103X.[133X
  
  
  [1X6.5-2 [33X[0;0YLeftInnerMappingGroup, RightInnerMappingGroup, MiddleInnerMappingGroup[133X[101X
  
  [29X[2XLeftInnerMappingGroup[102X( [3XQ[103X ) [32X attribute
  [29X[2XRightInnerMappingGroup[102X( [3XQ[103X ) [32X attribute
  [29X[2XMiddleInnerMappingGroup[102X( [3XQ[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10YThe  left  inner  mapping  group, right inner mapping group, resp.
            middle inner mapping group of a loop [3XQ[103X.[133X
  
  [1X6.5-3 InnerMappingGroup[101X
  
  [29X[2XInnerMappingGroup[102X( [3XQ[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10YThe inner mapping group of a loop [3XQ[103X.[133X
  
  [33X[0;0YHere is an example for multiplication groups and inner mapping groups:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XM := MoufangLoop(12,1);
[127X[104X
    [4X[28X<Moufang loop 12/1>
[128X[104X
    [4X[25Xgap>[125X [27XLeftSection(M)[2];
[127X[104X
    [4X[28X(1,2)(3,4)(5,6)(7,8)(9,12)(10,11)
[128X[104X
    [4X[25Xgap>[125X [27XMlt := MultiplicationGroup(M); Inn := InnerMappingGroup(M);
[127X[104X
    [4X[28X<permutation group of size 2592 with 23 generators>
[128X[104X
    [4X[28XGroup([ (4,6)(7,11), (7,11)(8,10), (2,6,4)(7,9,11), (3,5)(9,11), (8,12,10) ])
[128X[104X
    [4X[25Xgap>[125X [27XSize(Inn);
[127X[104X
    [4X[28X216
[128X[104X
  [4X[32X[104X
  
  
  [1X6.6 [33X[0;0YNuclei, Commutant, Center, and Associator Subloop[133X[101X
  
  [33X[0;0YSee Section [14X2.3[114X for the relevant definitions.[133X
  
  
  [1X6.6-1 [33X[0;0YLeftNucles, MiddleNucleus, and RightNucleus[133X[101X
  
  [29X[2XLeftNucleus[102X( [3XQ[103X ) [32X attribute
  [29X[2XMiddleNucleus[102X( [3XQ[103X ) [32X attribute
  [29X[2XRightNucleus[102X( [3XQ[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10YThe  left  nucleus,  middle  nucleus,  resp.  right  nucleus  of a
            quasigroup [3XQ[103X.[133X
  
  
  [1X6.6-2 [33X[0;0YNuc, NucleusOfQuasigroup and NucleusOfLoop[133X[101X
  
  [29X[2XNuc[102X( [3XQ[103X ) [32X attribute
  [29X[2XNucleusOfQuasigroup[102X( [3XQ[103X ) [32X attribute
  [29X[2XNucleusOfLoop[102X( [3XQ[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10YThese synonymous attributes return the nucleus of a quasigroup [3XQ[103X.[133X
  
  [33X[0;0YSince   all   nuclei   are   subquasigroups  of  [3XQ[103X,  they  are  returned  as
  subquasigroups  (resp.  subloops).  When  [3XQ[103X is a loop then all nuclei are in
  fact groups, and they are returned as associative loops.[133X
  
  [33X[0;0Y[12XRemark:[112X  The name [10XNucleus[110X is a global function of [5XGAP[105X with two variables. We
  have   therefore  used  [10XNuc[110X  rather  than  [10XNucleus[110X  for  the  nucleus.  This
  abbreviation is sometimes used in the literature, too.[133X
  
  [1X6.6-3 Commutant[101X
  
  [29X[2XCommutant[102X( [3XQ[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10YThe commutant of a quasigroup [3XQ[103X.[133X
  
  [1X6.6-4 Center[101X
  
  [29X[2XCenter[102X( [3XQ[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10YThe center of a quasigroup [3XQ[103X.[133X
  
  [33X[0;0YIf  [3XQ[103X is a loop, the center of [3XQ[103X is a subgroup of [3XQ[103X and it is returned as an
  associative loop.[133X
  
  [1X6.6-5 AssociatorSubloop[101X
  
  [29X[2XAssociatorSubloop[102X( [3XQ[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10YThe associator subloop of a loop [3XQ[103X.[133X
  
  [33X[0;0YWe calculate the associator subloop of [3XQ[103X as the smallest normal subloop of [3XQ[103X
  containing  all elements [22Xxbackslashα(x)[122X, where [22Xx[122X is an element of [3XQ[103X and [22Xα[122X is
  a left inner mapping of [3XQ[103X.[133X
  
  
  [1X6.7 [33X[0;0YNormal Subloops and Simple Loops[133X[101X
  
  [1X6.7-1 IsNormal[101X
  
  [29X[2XIsNormal[102X( [3XQ[103X, [3XS[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X if [3XS[103X is a normal subloop of a loop [3XQ[103X.[133X
  
  [33X[0;0YA  subloop  [22XS[122X  of  a  loop  [22XQ[122X  is  [13Xnormal[113X if it is invariant under all inner
  mappings of [22XQ[122X.[133X
  
  [1X6.7-2 NormalClosure[101X
  
  [29X[2XNormalClosure[102X( [3XQ[103X, [3XS[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10YThe normal closure of a subset [3XS[103X of a loop [3XQ[103X.[133X
  
  [33X[0;0YFor  a  subset  [22XS[122X  of a loop [22XQ[122X, the [13Xnormal closure[113X of [22XS[122X in [22XQ[122X is the smallest
  normal subloop of [22XQ[122X containing [22XS[122X.[133X
  
  [1X6.7-3 IsSimple[101X
  
  [29X[2XIsSimple[102X( [3XQ[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X if [3XQ[103X is a simple loop.[133X
  
  [33X[0;0YA loop [22XQ[122X is [13Xsimple[113X if [22X{1}[122X and [22XQ[122X are the only normal subloops of [22XQ[122X.[133X
  
  
  [1X6.8 [33X[0;0YFactor Loops[133X[101X
  
  [1X6.8-1 FactorLoop[101X
  
  [29X[2XFactorLoop[102X( [3XQ[103X, [3XS[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10YWhen  [3XS[103X  is  a normal subloop of a loop [3XQ[103X, returns the factor loop
            [3XQ[103X[22X/[122X[3XS[103X.[133X
  
  [1X6.8-2 NaturalHomomorphismByNormalSubloop[101X
  
  [29X[2XNaturalHomomorphismByNormalSubloop[102X( [3XQ[103X, [3XS[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10YWhen  [3XS[103X  is  a  normal  subloop  of  a loop [3XQ[103X, returns the natural
            projection from [3XQ[103X onto [3XQ[103X[22X/[122X[3XS[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XM := MoufangLoop( 12, 1 );; S := Subloop( M, [ M.3 ] );
[127X[104X
    [4X[28X<loop of order 3>
[128X[104X
    [4X[25Xgap>[125X [27XIsNormal( M, S );
[127X[104X
    [4X[28Xtrue
[128X[104X
    [4X[25Xgap>[125X [27XF := FactorLoop( M, S );
[127X[104X
    [4X[28X<loop of order 4>
[128X[104X
    [4X[25Xgap>[125X [27XNaturalHomomorphismByNormalSubloop( M, S );
[127X[104X
    [4X[28XMappingByFunction( <loop of order 12>, <loop of order 4>,
[128X[104X
    [4X[28X    function( x ) ... end )
[128X[104X
  [4X[32X[104X
  
  
  [1X6.9 [33X[0;0YNilpotency and Central Series[133X[101X
  
  [33X[0;0YSee Section [14X2.4[114X for the relevant definitions.[133X
  
  [1X6.9-1 IsNilpotent[101X
  
  [29X[2XIsNilpotent[102X( [3XQ[103X ) [32X property
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X if [3XQ[103X is a nilpotent loop.[133X
  
  [1X6.9-2 NilpotencyClassOfLoop[101X
  
  [29X[2XNilpotencyClassOfLoop[102X( [3XQ[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10YThe  nilpotency  class  of  a  loop  [3XQ[103X  if  [3XQ[103X  is  nilpotent, [10Xfail[110X
            otherwise.[133X
  
  [1X6.9-3 IsStronglyNilpotent[101X
  
  [29X[2XIsStronglyNilpotent[102X( [3XQ[103X ) [32X property
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X if [3XQ[103X is a strongly nilpotent loop.[133X
  
  [33X[0;0YA  loop  [22XQ[122X  is  said to be [13Xstrongly nilpotent[113X if its multiplication group is
  nilpotent.[133X
  
  [1X6.9-4 UpperCentralSeries[101X
  
  [29X[2XUpperCentralSeries[102X( [3XQ[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10YWhen [3XQ[103X is a nilpotent loop, returns the upper central series of [3XQ[103X,
            else returns [10Xfail[110X.[133X
  
  [1X6.9-5 LowerCentralSeries[101X
  
  [29X[2XLowerCentralSeries[102X( [3XQ[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10YWhen [3XQ[103X is a nilpotent loop, returns the lower central series of [3XQ[103X,
            else returns [10Xfail[110X.[133X
  
  [33X[0;0YThe [13Xlower central series[113X for loops is defined analogously to groups.[133X
  
  
  [1X6.10 [33X[0;0YSolvability, Derived Series and Frattini Subloop[133X[101X
  
  [33X[0;0YSee Section [14X2.4[114X for definitions of solvability an derived subloop.[133X
  
  [1X6.10-1 IsSolvable[101X
  
  [29X[2XIsSolvable[102X( [3XQ[103X ) [32X property
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X if [3XQ[103X is a solvable loop.[133X
  
  [1X6.10-2 DerivedSubloop[101X
  
  [29X[2XDerivedSubloop[102X( [3XQ[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10YThe derived subloop of a loop [3XQ[103X.[133X
  
  [1X6.10-3 DerivedLength[101X
  
  [29X[2XDerivedLength[102X( [3XQ[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10YIf  [3XQ[103X  is  solvable, returns the derived length of [3XQ[103X, else returns
            [10Xfail[110X.[133X
  
  
  [1X6.10-4 [33X[0;0YFrattiniSubloop and FrattinifactorSize[133X[101X
  
  [29X[2XFrattiniSubloop[102X( [3XQ[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10YThe  Frattini  subloop  of  [3XQ[103X.  The method is implemented only for
            strongly nilpotent loops.[133X
  
  [33X[0;0Y[13XFrattini subloop[113X of a loop [22XQ[122X is the intersection of maximal subloops of [22XQ[122X.[133X
  
  [1X6.10-5 FrattinifactorSize[101X
  
  [29X[2XFrattinifactorSize[102X( [3XQ[103X ) [32X attribute
  
  [1X6.11 [33X[0;0YIsomorphisms and Automorphisms[133X[101X
  
  [1X6.11-1 IsomorphismLoops[101X
  
  [29X[2XIsomorphismLoops[102X( [3XQ[103X, [3XL[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10YAn  isomorphism  from  a  loop  [3XQ[103X  to  a  loop  [3XL[103X if the loops are
            isomorphic, [10Xfail[110X otherwise.[133X
  
  [33X[0;0YIf  an  isomorphism exists, it is returned as a permutation [22Xf[122X of [22X1,dots,|[122X[3XQ[103X[22X|[122X,
  where  [22Xi^f=j[122X  means that the [22Xi[122Xth element of [3XQ[103X is mapped onto the [22Xj[122Xth element
  of  [3XL[103X. Note that this convention is used even if the underlying sets of [3XL[103X, [3XM[103X
  are not indexed by consecutive integers.[133X
  
  [1X6.11-2 LoopsUpToIsomorphism[101X
  
  [29X[2XLoopsUpToIsomorphism[102X( [3Xls[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10YGiven  a  list  [3Xls[103X of loops, returns a sublist of [3Xls[103X consisting of
            representatives of isomorphism classes of loops from [3Xls[103X.[133X
  
  [1X6.11-3 AutomorphismGroup[101X
  
  [29X[2XAutomorphismGroup[102X( [3XQ[103X ) [32X attribute
  [6XReturns:[106X  [33X[0;10YThe  automorphism  group  of a loop [3XQ[103X, with the same convention on
            permutations as in the case of [10XIsomorphismLoops[110X.[133X
  
  [33X[0;0Y[12XRemark:[112X  Since  two isomorphisms differ by an automorphism, all isomorphisms
  from  [3XQ[103X  to  [3XL[103X can be obtained by a combination of [10XIsomorphismLoops([3XQ[103X[10X,[3XL[103X[10X)[110X and
  [10XAutomorphismGroup([3XL[103X[10X)[110X.[133X
  
  [33X[0;0YWhile  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. [5XLOOPS[105X contains several functions for this purpose.[133X
  
  [1X6.11-4 IsomorphicCopyByPerm[101X
  
  [29X[2XIsomorphicCopyByPerm[102X( [3XQ[103X, [3Xf[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10YWhen  [3XQ[103X  is  a  quasigroup  and  [3Xf[103X is a permutation of [22X1,dots,|[122X[3XQ[103X[22X|[122X,
            returns   a   quasigroup  defined  on  the  same  set  as  [3XQ[103X  with
            multiplication  [22X*[122X  defined  by [22Xx*y =[122X[3Xf[103X[22X([122X[3Xf[103X[22X^-1(x)[122X[3Xf[103X[22X^-1(y))[122X. When [3XQ[103X is a
            declared  loop,  a  loop  is  returned.  Consequently, when [3XQ[103X is a
            declared  loop  and [3Xf[103X[22X(1) = kne 1[122X, then [3Xf[103X is first replaced with [3Xf[103X[22X∘
            (1,k)[122X, to make sure that the resulting Cayley table is normalized.[133X
  
  [1X6.11-5 IsomorphicCopyByNormalSubloop[101X
  
  [29X[2XIsomorphicCopyByNormalSubloop[102X( [3XQ[103X, [3XS[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10YWhen [3XS[103X is a normal subloop of a loop [3XQ[103X, returns an isomorphic copy
            of  [3XQ[103X  in  which  the  elements are ordered according to the right
            cosets  of  [3XS[103X. In particular, the Cayley table of [3XS[103X will appear in
            the top left corner of the Cayley table of the resulting loop.[133X
  
  [33X[0;0YIn 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.[133X
  
  [1X6.11-6 Discriminator[101X
  
  [29X[2XDiscriminator[102X( [3XQ[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10YA  data  structure  with  isomorphism  invariants of a loop [3XQ[103X. See
            [Voj06] or the file [11Xiso.gi[111X for more details.[133X
  
  [33X[0;0YIf two loops have different discriminators, they are not isomorphic. If they
  have identical discriminators, they may or may not be isomorphic.[133X
  
  [1X6.11-7 AreEqualDiscriminators[101X
  
  [29X[2XAreEqualDiscriminators[102X( [3XD1[103X, [3XD2[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Y[10Xtrue[110X  if  [3XD1[103X,  [3XD2[103X  are  equal  discriminators  for the purposes of
            isomorphism searches.[133X
  
  
  [1X6.12 [33X[0;0YIsotopisms[133X[101X
  
  [33X[0;0YAt the moment, [5XLOOPS[105X contains only slow methods for testing if two loops are
  isotopic.  The method works as follows: It is well known that if a loop [22XK[122X is
  isotopic  to  a loop [22XL[122X then there exist a principal loop isotope [22XP[122X of [22XK[122X such
  that  [22XP[122X is isomorphic to [22XL[122X. The algorithm first finds all principal isotopes
  of [22XK[122X, then filters them up to isomorphism, and then checks if any of them is
  isomorphic to [22XL[122X. This is rather slow already for small orders.[133X
  
  [1X6.12-1 IsotopismLoops[101X
  
  [29X[2XIsotopismLoops[102X( [3XK[103X, [3XL[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10Y[10Xfail[110X  if [3XK[103X, [3XL[103X are not isotopic loops, else it returns an isotopism
            as a triple of bijections on [22X1,dots,|[122X[3XK[103X[22X|[122X.[133X
  
  [1X6.12-2 LoopsUpToIsotopism[101X
  
  [29X[2XLoopsUpToIsotopism[102X( [3Xls[103X ) [32X operation
  [6XReturns:[106X  [33X[0;10YGiven  a  list  [3Xls[103X of loops, returns a sublist of [3Xls[103X consisting of
            representatives of isotopism classes of loops from [3Xls[103X.[133X
  
