  
  [1X4 [33X[0;0YIsoclinism of groups and crossed modules[133X[101X
  
  [33X[0;0YThis  chapter  describes  some  functions  written by Alper Odabaş and Enver
  Uslu,  and  reported  in  their  paper  [IOU16].  Section  [14X4.1[114X contains some
  additional  basic  functions  for  crossed  modules, constructing quotients,
  centres,  centralizers  and  normalizers.  In Sections [14X4.2[114X and [14X4.3[114X there are
  functions  dealing  specifically  with isoclinism for groups and for crossed
  modules.  Since  these  functions represent a recent addition to the package
  (as  of  November  2015),  the function names are liable to change in future
  versions.  The  notion  of isoclinism has been crucial to the enumeration of
  groups  of  prime  power  order,  see for example James, Newman and O'Brien,
  [JNO90].[133X
  
  
  [1X4.1 [33X[0;0YMore operations for crossed modules[133X[101X
  
  [1X4.1-1 FactorPreXMod[101X
  
  [33X[1;0Y[29X[2XFactorPreXMod[102X( [3XX1[103X, [3XX2[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XNaturalMorphismByNormalSubPreXMod[102X( [3XX1[103X, [3XX2[103X ) [32X operation[133X
  
  [33X[0;0YWhen [22XcalX_2 = (∂_2 : S_2 -> R_2)[122X is a normal sub-precrossed module of [22XcalX_1
  = (∂_1 : S_1 -> R_1)[122X, then the quotient precrossed module is [22X(∂ : S_2/S_1 ->
  R_2/R_1)[122X with the induced boundary and action maps. Quotienting a precrossed
  module  by  it's  Peiffer  subgroup  is a special case of this construction.
  (Permutation representations vary between different versions of [5XGAP[105X.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xd24 := DihedralGroup( IsPermGroup, 24 );; [127X[104X
    [4X[25Xgap>[125X [27XSetName( d24, "d24" );[127X[104X
    [4X[25Xgap>[125X [27XY24 := XModByAutomorphismGroup( d24 );; [127X[104X
    [4X[25Xgap>[125X [27XSize( Y24 );[127X[104X
    [4X[28X[ 24, 48 ][128X[104X
    [4X[25Xgap>[125X [27XX24 := Image( IsomorphismPerm2DimensionalGroup( Y24 ) );[127X[104X
    [4X[28X[d24->Group([ (2,4), (1,2,3,4), (6,7), (5,6,7) ])][128X[104X
    [4X[25Xgap>[125X [27Xnsx := NormalSubXMods( X24 );; [127X[104X
    [4X[25Xgap>[125X [27XLength( nsx );[127X[104X
    [4X[28X40[128X[104X
    [4X[25Xgap>[125X [27Xids := List( nsx, n -> IdGroup(n) );; [127X[104X
    [4X[25Xgap>[125X [27Xpos1 := Position( ids, [ [4,1], [8,3] ] );;[127X[104X
    [4X[25Xgap>[125X [27XXn1 := nsx[pos1]; [127X[104X
    [4X[28X[Group( [ f2*f4^2, f3*f4 ] )->Group( [ f3, f4, f5 ] )][128X[104X
    [4X[25Xgap>[125X [27Xnat1 := NaturalMorphismByNormalSubPreXMod( X24, Xn1 );; [127X[104X
    [4X[25Xgap>[125X [27XQn1 := FactorPreXMod( X24, Xn1 );; [127X[104X
    [4X[25Xgap>[125X [27X[ Size( Xn1 ), Size( Qn1 ) ];[127X[104X
    [4X[28X[ [ 4, 8 ], [ 6, 6 ] ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X4.1-2 IntersectionSubXMods[101X
  
  [33X[1;0Y[29X[2XIntersectionSubXMods[102X( [3XX0[103X, [3XX1[103X, [3XX2[103X ) [32X operation[133X
  
  [33X[0;0YWhen  [10XX1,X2[110X are subcrossed modules of [10XX0[110X, then the source and range of their
  intersection  are  the  intersections of the sources and ranges of [10XX1[110X and [10XX2[110X
  respectively.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xpos2 := Position( ids, [ [24,6], [12,4] ] );;[127X[104X
    [4X[25Xgap>[125X [27XXn2 := nsx[pos2];; [127X[104X
    [4X[25Xgap>[125X [27XIdGroup( Xn2 );         [127X[104X
    [4X[28X[ [ 24, 6 ], [ 12, 4 ] ][128X[104X
    [4X[25Xgap>[125X [27Xpos3 := Position( ids, [ [12,2], [24,5] ] );;[127X[104X
    [4X[25Xgap>[125X [27XXn3 := nsx[pos3];; [127X[104X
    [4X[25Xgap>[125X [27XIdGroup( Xn3 );[127X[104X
    [4X[28X[ [ 12, 2 ], [ 24, 5 ] ][128X[104X
    [4X[25Xgap>[125X [27XXn23 := IntersectionSubXMods( X24, Xn2, Xn3 );;[127X[104X
    [4X[25Xgap>[125X [27XIdGroup( Xn23 );[127X[104X
    [4X[28X[ [ 12, 2 ], [ 6, 2 ] ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X4.1-3 Displacement[101X
  
  [33X[1;0Y[29X[2XDisplacement[102X( [3Xalpha[103X, [3Xr[103X, [3Xs[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XDisplacementGroup[102X( [3XX0[103X, [3XQ[103X, [3XT[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XDisplacementSubgroup[102X( [3XX0[103X ) [32X attribute[133X
  
  [33X[0;0YCommutators  may  be  written  [22X[r,q] = r^-1q^-1rq = (q^-1)^rq = r^-1r^q[122X, and
  satisfy identities[133X
  
  
  [24X[33X[0;6Y[r,q]^p   =  [r^p,q^p],  \qquad  [pr,q]  =  [p,q]^r[r,q],  \qquad  [r,pq]  =
  [r,q][r,p]^q, \qquad [r,q]^{-1} = [q,r].[133X
  
  [124X
  
  [33X[0;0YIn  a similar way, when a group [22XR[122X acts on a group [22XS[122X, the [13Xdisplacement[113X of [22Xs ∈
  S[122X  by [22Xr ∈ R[122X is defined to be [22X⟨ r,s ⟩ := (s^-1)^rs ∈ S[122X. When [22XcalX = (∂ : S ->
  R)[122X is a pre-crossed module, the first crossed module axiom requires [22X∂⟨ r,s ⟩
  =  [r,∂  s][122X.  When [22Xα[122X is the action of [22XcalX[122X, the [2XDisplacement[102X function may be
  used  to  calculate [22X⟨ r,s ⟩[122X. Displacements satisfy the following identities,
  where [22Xs,t ∈ S,~ p,q,r ∈ R[122X:[133X
  
  
  [24X[33X[0;6Y\langle r,s \rangle^p = \langle r^p,s^p \rangle, \qquad \langle qr,s \rangle
  =  \langle  q,s \rangle^r \langle r,s \rangle, \qquad \langle r,st \rangle =
  \langle r,t \rangle \langle r,s \rangle^t, \qquad \langle r,s \rangle^{-1} =
  \langle r^{-1},s^r \rangle.[133X
  
  [124X
  
  [33X[0;0YThe  operation  [2XDisplacementGroup[102X  applied  to  [10XX0,Q,T[110X  is the subgroup of [10XS[110X
  consisting  of  all  the  displacements  [22X⟨  r,s  ⟩,  r ∈ Q leqslant R, s ∈ T
  leqslant S[122X. The [2XDisplacementSubgroup[102X of [22XcalX[122X is the subgroup [22XDisp(calX)[122X of [22XS[122X
  given  by  [10XDisplacementGroup(X0,R,S)[110X.  The  identities  imply  [22X⟨ r,s ⟩^t = ⟨
  r,st^r^-1} ⟩ ⟨ r^-1,t ⟩[122X, so [22XDisp(calX)[122X is normal in [22XS[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xpos4 := Position( ids, [ [6,2], [24,14] ] );;[127X[104X
    [4X[25Xgap>[125X [27XXn4 := nsx[pos4];; [127X[104X
    [4X[25Xgap>[125X [27Xbn4 := Boundary( Xn4 );;[127X[104X
    [4X[25Xgap>[125X [27XSn4 := Source(Xn4);; [127X[104X
    [4X[25Xgap>[125X [27XRn4 := Range(Xn4);; [127X[104X
    [4X[25Xgap>[125X [27XgenRn4 := GeneratorsOfGroup( Rn4 );;[127X[104X
    [4X[25Xgap>[125X [27XL := List( genRn4, g -> ( Order(g) = 2 ) and [127X[104X
    [4X[25X>[125X [27X                not ( IsNormal( Rn4, Subgroup( Rn4, [g] ) ) ) );;[127X[104X
    [4X[25Xgap>[125X [27Xpos := Position( L, true );;[127X[104X
    [4X[25Xgap>[125X [27Xs := Sn4.1;  r := genRn4[pos]; [127X[104X
    [4X[28X(1,3,5,7,9,11)(2,4,6,8,10,12)[128X[104X
    [4X[28X(6,7)[128X[104X
    [4X[25Xgap>[125X [27Xact := XModAction( Xn4 );; [127X[104X
    [4X[25Xgap>[125X [27Xd := Displacement( act, r, s );[127X[104X
    [4X[28X(1,5,9)(2,6,10)(3,7,11)(4,8,12)[128X[104X
    [4X[25Xgap>[125X [27XImage( bn4, d ) = Comm( r, Image( bn4, s ) );  [127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XQn4 := Subgroup( Rn4, [ (6,7), (1,3), (2,4) ] );;   [127X[104X
    [4X[25Xgap>[125X [27XTn4 := Subgroup( Sn4, [ (1,3,5,7,9,11)(2,4,6,8,10,12) ] );;[127X[104X
    [4X[25Xgap>[125X [27XDisplacementGroup( Xn4, Qn4, Tn4 );                        [127X[104X
    [4X[28XGroup([ (1,5,9)(2,6,10)(3,7,11)(4,8,12) ])[128X[104X
    [4X[25Xgap>[125X [27XDisplacementSubgroup( Xn4 );[127X[104X
    [4X[28XGroup([ (1,5,9)(2,6,10)(3,7,11)(4,8,12) ])[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X4.1-4 CommutatorSubXMod[101X
  
  [33X[1;0Y[29X[2XCommutatorSubXMod[102X( [3XX[103X, [3XX1[103X, [3XX2[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XCrossActionSubgroup[102X( [3XX[103X, [3XX1[103X, [3XX2[103X ) [32X operation[133X
  
  [33X[0;0YWhen  [22XcalX_1 = (N -> Q), calX_2 = (M -> P)[122X are two normal subcrossed modules
  of  [22XcalX  = (∂ : S -> R)[122X, the displacements [22X⟨ p,n ⟩[122X and [22X⟨ q,m ⟩[122X all map by [22X∂[122X
  into  [22X[Q,P][122X.  These  displacements  form  a normal subgroup of [22XS[122X, called the
  [10XCrossActionSubgroup[110X. The [10XCommutatorSubXMod[110X [22X[calX_1,calX_2][122X has this subgroup
  as source and [22X[P,Q][122X as range, and is normal in [22XcalX[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XCAn23 := CrossActionSubgroup( X24, Xn2, Xn3 );;[127X[104X
    [4X[25Xgap>[125X [27XIdGroup( CAn23 );[127X[104X
    [4X[28X[ 12, 2 ][128X[104X
    [4X[25Xgap>[125X [27XCn23 := CommutatorSubXMod( X24, Xn2, Xn3 );;[127X[104X
    [4X[25Xgap>[125X [27XIdGroup( Cn23 );[127X[104X
    [4X[28X[ [ 12, 2 ], [ 6, 2 ] ][128X[104X
    [4X[25Xgap>[125X [27XXn23 = Cn23;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X4.1-5 DerivedSubXMod[101X
  
  [33X[1;0Y[29X[2XDerivedSubXMod[102X( [3XX0[103X ) [32X attribute[133X
  
  [33X[0;0YThe [10XDerivedSubXMod[110X of [22XcalX[122X is the normal subcrossed module [22X[calX,calX] = (∂'
  :  Disp(calX)  ->  [R,R])[122X  where  [22X∂'[122X is the restriction of [22X∂[122X (see page 66 of
  Norrie's thesis [Nor87]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XDXn4 := DerivedSubXMod( Xn4 );;[127X[104X
    [4X[25Xgap>[125X [27XIdGroup( DXn4 );[127X[104X
    [4X[28X[ [ 3, 1 ], [ 3, 1 ] ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X4.1-6 FixedPointSubgroupXMod[101X
  
  [33X[1;0Y[29X[2XFixedPointSubgroupXMod[102X( [3XX0[103X, [3XT[103X, [3XQ[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XStabilizerSubgroupXMod[102X( [3XX0[103X, [3XT[103X, [3XQ[103X ) [32X operation[133X
  
  [33X[0;0YThe  [10XFixedPointSubgroupXMod(X,T,Q)[110X  for  [22XcalX=(∂  :  S -> R)[122X is the subgroup
  [22XFix(calX,T,Q)[122X  of  elements  [22Xt  ∈  T leqslant S[122X individually fixed under the
  action of [22XQ leqslant R[122X.[133X
  
  [33X[0;0YThe [10XStabilizerSubgroupXMod(X,T,Q)[110X for [22XcalX[122X is the subgroup [22XStab(calX,T,Q)[122X of
  [22XQ  leqslant R[122X whose elements act trivially on the whole of [22XT leqslant S[122X (see
  page 19 of Norrie's thesis [Nor87]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xfix := FixedPointSubgroupXMod( Xn4, Sn4, Rn4 );[127X[104X
    [4X[28XGroup([ (1,7)(2,8)(3,9)(4,10)(5,11)(6,12) ])[128X[104X
    [4X[25Xgap>[125X [27Xstab := StabilizerSubgroupXMod( Xn4, Sn4, Rn4 );;[127X[104X
    [4X[25Xgap>[125X [27XIdGroup( stab );[127X[104X
    [4X[28X[ 12, 5 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X4.1-7 CentreXMod[101X
  
  [33X[1;0Y[29X[2XCentreXMod[102X( [3XX0[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XCentralizer[102X( [3XX[103X, [3XY[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XNormalizer[102X( [3XX[103X, [3XY[103X ) [32X operation[133X
  
  [33X[0;0YThe  [13Xcentre[113X  [22XZ(calX)[122X  of  [22XcalX  = (∂ : S -> R)[122X has as source the fixed point
  subgroup  [22XFix(calX,S,R)[122X.  The  range  is the intersection of the centre [22XZ(R)[122X
  with the stabilizer subgroup.[133X
  
  [33X[0;0YWhen  [22XcalY  =  (T  ->  Q)[122X is a subcrossed module of [22XcalX = (∂ : S -> R)[122X, the
  [13Xcentralizer[113X  [22XC_calX(calY)[122X  of  [22XcalY[122X  has  as source the fixed point subgroup
  [22XFix(calX,S,Q)[122X.  The range is the intersection of the centralizer [22XC_R(Q)[122X with
  [22XStab(calX,T,R)[122X.[133X
  
  [33X[0;0YThe  [13Xnormalizer[113X  [22XN_calX(calY)[122X  of  [22XcalY[122X  has  as  source  the  subgroup of [22XS[122X
  consisting of the displacements [22X⟨ s,q ⟩[122X which lie in [22XS[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XZXn4 := CentreXMod( Xn4 );      [127X[104X
    [4X[28X[Group( [ f3*f4 ] )->Group( [ f3, f5 ] )][128X[104X
    [4X[25Xgap>[125X [27XIdGroup( ZXn4 );[127X[104X
    [4X[28X[ [ 2, 1 ], [ 4, 2 ] ][128X[104X
    [4X[25Xgap>[125X [27XCDXn4 := Centralizer( Xn4, DXn4 );[127X[104X
    [4X[28X[Group( [ f3*f4 ] )->Group( [ f2 ] )][128X[104X
    [4X[25Xgap>[125X [27XIdGroup( CDXn4 );    [127X[104X
    [4X[28X[ [ 2, 1 ], [ 3, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27XNDXn4 := Normalizer( Xn4, DXn4 ); [127X[104X
    [4X[28X[Group( <identity> of ... )->Group( [ f5, f2*f3 ] )][128X[104X
    [4X[25Xgap>[125X [27XIdGroup( NDXn4 );[127X[104X
    [4X[28X[ [ 1, 1 ], [ 12, 5 ] ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X4.1-8 CentralQuotient[101X
  
  [33X[1;0Y[29X[2XCentralQuotient[102X( [3XG[103X ) [32X attribute[133X
  
  [33X[0;0YThe  [10XCentralQuotient[110X  of  a group [22XG[122X is the crossed module [22X(G -> G/Z(G))[122X with
  the  natural  homomorphism  as  the  boundary map. This is a special case of
  [2XXModByCentralExtension[102X ([14X2.1-5[114X).[133X
  
  [33X[0;0YSimilarly,  the  central  quotient  of  a crossed module [22XcalX[122X is the crossed
  square [22X(calX ⇒ calX/Z(calX))[122X (see section [14X8.2[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XQ24 := CentralQuotient( d24);  IdGroup( Q24 );                     [127X[104X
    [4X[28X[d24->d24/Z(d24)][128X[104X
    [4X[28X[ [ 24, 6 ], [ 12, 4 ] ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X4.1-9 IsAbelian2DimensionalGroup[101X
  
  [33X[1;0Y[29X[2XIsAbelian2DimensionalGroup[102X( [3XX0[103X ) [32X property[133X
  [33X[1;0Y[29X[2XIsAspherical2DimensionalGroup[102X( [3XX0[103X ) [32X property[133X
  [33X[1;0Y[29X[2XIsSimplyConnected2DimensionalGroup[102X( [3XX0[103X ) [32X property[133X
  [33X[1;0Y[29X[2XIsFaithful2DimensionalGroup[102X( [3XX0[103X ) [32X property[133X
  
  [33X[0;0YA crossed module is [13Xabelian[113X if it equal to its centre. This is the case when
  the range group is abelian and the action is trivial.[133X
  
  [33X[0;0YA crossed module is [13Xaspherical[113X if the boundary has trivial kernel.[133X
  
  [33X[0;0YA crossed module is [13Xsimply connected[113X if the boundary has trivial cokernel.[133X
  
  [33X[0;0YA crossed module is [13Xfaithful[113X if the action is faithful.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27X[ IsAbelian2DimensionalGroup(Xn4), IsAbelian2DimensionalGroup(X24) ];
[127X[104X
    [4X[28X[ false, false ][128X[104X
    [4X[25Xgap>[125X [27Xpos7 := Position( ids, [ [3,1], [6,1] ] );;
[127X[104X
    [4X[25Xgap>[125X [27X[ IsAspherical2DimensionalGroup(nsx[pos7]), IsAspherical2DimensionalGroup(X24) ];
[127X[104X
    [4X[28X[ true, false ]
[128X[104X
    [4X[25Xgap>[125X [27X[ IsSimplyConnected2DimensionalGroup(Xn4), IsSimplyConnected2DimensionalGroup(X24) ];[127X[104X
    [4X[28X[ true, true ][128X[104X
    [4X[25Xgap>[125X [27X[ IsFaithful2DimensionalGroup(Xn4), IsFaithful2DimensionalGroup(X24) ];              [127X[104X
    [4X[28X[ false, true ] [128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X4.1-10 LowerCentralSeriesOfXMod[101X
  
  [33X[1;0Y[29X[2XLowerCentralSeriesOfXMod[102X( [3XX0[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XIsNilpotent2DimensionalGroup[102X( [3XX0[103X ) [32X property[133X
  [33X[1;0Y[29X[2XNilpotencyClass2DimensionalGroup[102X( [3XX0[103X ) [32X attribute[133X
  
  [33X[0;0YLet  [22XcalY[122X be a subcrossed module of [22XcalX[122X. A [13X series of length[113X [22Xn[122X from [22XcalX[122X to
  [22XcalY[122X has the form[133X
  
  
  [24X[33X[0;6Y\calX ~=~ \calX_0 ~\unrhd~ \calX_1 ~\unrhd~ \cdots ~\unrhd~ \calX_i ~\unrhd~
  \cdots ~\unrhd~ \calX_n ~=~ \calY \quad (1 \leqslant i \leqslant n).[133X
  
  [124X
  
  [33X[0;0YThe quotients [22XcalF_i = calX_i / calX_i-1[122X are the [13Xfactors[113X of the series.[133X
  
  [33X[0;0YA  factor  [22XcalF_i[122X  is  [13Xcentral[113X if [22XcalX_i-1 ⊴ calX[122X and [22XcalF_i[122X is a subcrossed
  module of the centre of [22XcalX / calX_i-1[122X.[133X
  
  [33X[0;0YA series is [13Xcentral[113X if all its factors are central.[133X
  
  [33X[0;0Y[22XcalX[122X is [13Xsoluble[113X if it has a series all of whose factors are abelian.[133X
  
  [33X[0;0Y[22XcalX[122X  is  [13Xnilpotent[113X  is  it  has  a  series all of whose factors are central
  factors of [22XcalX[122X.[133X
  
  [33X[0;0YThe [13Xlower central series[113X of [22XcalX[122X is the sequence (see [Nor87], p.77):[133X
  
  
  [24X[33X[0;6Y\calX  ~=~  \Gamma_1(\calX)  ~\unrhd~ \Gamma_2(\calX) ~\unrhd~ \cdots \qquad
  \mbox{where} \qquad \Gamma_j(\calX) ~=~ [ \Gamma_{j-1}(\calX), \calX].[133X
  
  [124X
  
  [33X[0;0YIf  [22XcalX[122X  is  nilpotent,  then  its lower central series is its most rapidly
  descending central series.[133X
  
  [33X[0;0YThe  least  integer [22Xc[122X such that [22XΓ_c+1(calX)[122X is the trivial crossed module is
  the [13Xnilpotency class[113X of [22XcalX[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xlcs := LowerCentralSeries( X24 );;      [127X[104X
    [4X[25Xgap>[125X [27XList( lcs, g -> IdGroup(g) );[127X[104X
    [4X[28X[ [ [ 24, 6 ], [ 48, 38 ] ], [ [ 12, 2 ], [ 6, 2 ] ], [ [ 6, 2 ], [ 3, 1 ] ], [128X[104X
    [4X[28X  [ [ 3, 1 ], [ 3, 1 ] ] ][128X[104X
    [4X[25Xgap>[125X [27XIsNilpotent2DimensionalGroup( X24 );      [127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XNilpotencyClassOf2DimensionalGroup( X24 );[127X[104X
    [4X[28X0[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X4.1-11 IsomorphismXMods[101X
  
  [33X[1;0Y[29X[2XIsomorphismXMods[102X( [3XX1[103X, [3XX2[103X ) [32X operation[133X
  
  [33X[0;0YThe  function [10XIsomorphismXMods[110X computes an isomorphism [22Xμ : calX_1 -> calX_2[122X,
  provided one exists, or else returns [10Xfail[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xgend24 := GeneratorsOfGroup( d24 );;             [127X[104X
    [4X[25Xgap>[125X [27Xa := gend24[1];; b:= gend24[2];; [127X[104X
    [4X[25Xgap>[125X [27XJ := Subgroup( d24, [a^2,b] ); [127X[104X
    [4X[28XGroup([ (1,3,5,7,9,11)(2,4,6,8,10,12), (2,12)(3,11)(4,10)(5,9)(6,8) ])[128X[104X
    [4X[25Xgap>[125X [27XK := Subgroup( d24, [a^2,a*b] );[127X[104X
    [4X[28XGroup([ (1,3,5,7,9,11)(2,4,6,8,10,12), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7) ])[128X[104X
    [4X[25Xgap>[125X [27XXJ := XModByNormalSubgroup( d24, J );;[127X[104X
    [4X[25Xgap>[125X [27XXK := XModByNormalSubgroup( d24, K );;[127X[104X
    [4X[25Xgap>[125X [27Xiso := IsomorphismXMods( XJ, XK );;[127X[104X
    [4X[25Xgap>[125X [27XSourceHom( iso );[127X[104X
    [4X[28X[ (1,3,5,7,9,11)(2,4,6,8,10,12), (2,12)(3,11)(4,10)(5,9)(6,8) ] -> [128X[104X
    [4X[28X[ (1,3,5,7,9,11)(2,4,6,8,10,12), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7) ][128X[104X
    [4X[25Xgap>[125X [27XRangeHom( iso ); [127X[104X
    [4X[28X[ (1,2,3,4,5,6,7,8,9,10,11,12), (2,12)(3,11)(4,10)(5,9)(6,8) ] -> [128X[104X
    [4X[28X[ (1,2,3,4,5,6,7,8,9,10,11,12), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7) ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X4.1-12 AllXMods[101X
  
  [33X[1;0Y[29X[2XAllXMods[102X( [3Xargs[103X ) [32X function[133X
  [33X[1;0Y[29X[2XAllXModsWithGroups[102X( [3Xsrc[103X, [3Xrng[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XAllXModsUpToIsomorphism[102X( [3Xsrc[103X, [3Xrng[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XIsomorphismClassRepresentatives2dGroups[102X( [3XL[103X ) [32X operation[133X
  
  [33X[0;0YThe  global  function  [10XAllXMods[110X  may  be  called  in three ways. Firstly, as
  [10XAllXMods(S,R)[110X  to  compute  all crossed modules with chosen source and range
  groups:   this   calls  [10XAllXModsWithGroups(S,R)[110X.  Secondly,  [10XAllXMods([n,m])[110X
  computes  all crossed modules with a given size [10X[n,m][110X. Thirdly [10XAllXMods(ord)[110X
  to  compute  all  crossed  modules whose associated cat1-groups have a given
  size [10Xord[110X.[133X
  
  [33X[0;0YThe  function [10XAllXModsUpToIsomorphism(S,R)[110X returns a list of representatives
  of the isomorphism classes of crossed modules with source [22XS[122X and range [22XR[122X.[133X
  
  [33X[0;0YIf  [10XL[110X  is a list returned by, for example, [10XAllXModsWithGroups(S,R)[110X, then the
  isomorphism   class   representatives   for   this   list   is  returned  by
  [10XIsomorphismClassRepresentatives2dGroups(L)[110X.  This result is the same as that
  given by [10XAllXModsUpToIsomorphism(S,R)[110X.[133X
  
  [33X[0;0YIn  the  example  we  see that there are [22X4[122X crossed modules, in [22X3[122X isomorphism
  classes,  [22X(C_6 -> S_3)[122X; forming a subset of the [22X17[122X crossed modules with size
  [10X[6,6][110X;  and  that  these  form  a  subset  of  the [22X205[122X crossed modules whose
  cat1-group has size [22X36[122X. There are [22X40[122X precrossed modules with size [10X[6,6][110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xc6 := SmallGroup( 6, 2 );; [127X[104X
    [4X[25Xgap>[125X [27Xs3 := SmallGroup( 6, 1 );; [127X[104X
    [4X[25Xgap>[125X [27XAc6s3 := AllXMods( c6, s3 );;   [127X[104X
    [4X[25Xgap>[125X [27XLength( Ac6s3 );           [127X[104X
    [4X[28X4[128X[104X
    [4X[25Xgap>[125X [27XIc6s3 := AllXModsUpToIsomorphism( c6, s3 );; [127X[104X
    [4X[25Xgap>[125X [27XList( Ic6s3, obj -> IsTrivialAction2DimensionalGroup( obj ) ); [127X[104X
    [4X[28X[ true, false, false ][128X[104X
    [4X[25Xgap>[125X [27XKc6s3 := List( Ic6s3, obj -> KernelCokernelXMod( obj ) );; [127X[104X
    [4X[25Xgap>[125X [27XList( Kc6s3, obj -> IdGroup( obj ) );[127X[104X
    [4X[28X[ [ [ 6, 2 ], [ 6, 1 ] ], [ [ 6, 2 ], [ 6, 1 ] ], [ [ 2, 1 ], [ 2, 1 ] ] ][128X[104X
    [4X[28X[ ][128X[104X
    [4X[25Xgap>[125X [27XA66 := AllXMods( [6,6] );;   [127X[104X
    [4X[25Xgap>[125X [27XLength( A66 );[127X[104X
    [4X[28X17[128X[104X
    [4X[25Xgap>[125X [27XIA66 := IsomorphismClassRepresentatives2dGroups( A66 );;[127X[104X
    [4X[25Xgap>[125X [27XLength( IA66 );[127X[104X
    [4X[28X9[128X[104X
    [4X[25Xgap>[125X [27Xx36 := AllXMods( 36 );; [127X[104X
    [4X[25Xgap>[125X [27XLength( x36 ); [127X[104X
    [4X[28X205[128X[104X
    [4X[25Xgap>[125X [27Xsize36 := List( x36, x -> [ Size(Source(x)), Size(Range(x)) ] );;[127X[104X
    [4X[25Xgap>[125X [27XCollected( size36 );[127X[104X
    [4X[28X[ [ [ 1, 36 ], 14 ], [ [ 2, 18 ], 7 ], [ [ 3, 12 ], 21 ], [ [ 4, 9 ], 14 ], [128X[104X
    [4X[28X  [ [ 6, 6 ], 17 ], [ [ 9, 4 ], 102 ], [ [ 12, 3 ], 8 ], [ [ 18, 2 ], 18 ], [128X[104X
    [4X[28X  [ [ 36, 1 ], 4 ] ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X4.2 [33X[0;0YIsoclinism for groups[133X[101X
  
  [1X4.2-1 Isoclinism[101X
  
  [33X[1;0Y[29X[2XIsoclinism[102X( [3XG[103X, [3XH[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XAreIsoclinicDomains[102X( [3XG[103X, [3XH[103X ) [32X operation[133X
  
  [33X[0;0YLet [22XG,H[122X be groups with central quotients [22XQ(G)[122X and [22XQ(H)[122X and derived subgroups
  [22X[G,G][122X  and  [22X[H,H][122X respectively. Let [22Xc_G : G/Z(G) × G/Z(G) -> [G,G][122X and [22Xc_H :
  H/Z(H)  × H/Z(H) -> [H,H][122X be the two commutator maps. An [13Xisoclinism[113X [22XG ∼ H[122X is
  a  pair  of  isomorphisms  [22X(η,ξ)[122X where [22Xη : G/Z(G) -> H/Z(H)[122X and [22Xξ : [G,G] ->
  [H,H][122X  such  that  [22Xc_G  *  ξ  =  (η × η) * c_H[122X. Isoclinism is an equivalence
  relation, and all abelian groups are isoclinic to the trivial group.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XG := SmallGroup( 64, 6 );;  StructureDescription( G ); [127X[104X
    [4X[28X"(C8 x C4) : C2"[128X[104X
    [4X[25Xgap>[125X [27XQG := CentralQuotient( G );;  IdGroup( QG );[127X[104X
    [4X[28X[ [ 64, 6 ], [ 8, 3 ] ][128X[104X
    [4X[25Xgap>[125X [27XH := SmallGroup( 32, 41 );;  StructureDescription( H );[127X[104X
    [4X[28X"C2 x Q16"[128X[104X
    [4X[25Xgap>[125X [27XQH := CentralQuotient( H );;  IdGroup( QH );[127X[104X
    [4X[28X[ [ 32, 41 ], [ 8, 3 ] ][128X[104X
    [4X[25Xgap>[125X [27XIsoclinism( G, H );[127X[104X
    [4X[28X[ [ f1, f2, f3 ] -> [ f1, f2*f3, f3 ], [ f3, f5 ] -> [ f4*f5, f5 ] ][128X[104X
    [4X[25Xgap>[125X [27XK := SmallGroup( 32, 43 );;  StructureDescription( K );[127X[104X
    [4X[28X"(C2 x D8) : C2"[128X[104X
    [4X[25Xgap>[125X [27XQK := CentralQuotient( K );;  IdGroup( QK );                       [127X[104X
    [4X[28X[ [ 32, 43 ], [ 16, 11 ] ][128X[104X
    [4X[25Xgap>[125X [27XAreIsoclinicDomains( G, K );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X4.2-2 IsStemDomain[101X
  
  [33X[1;0Y[29X[2XIsStemDomain[102X( [3XG[103X ) [32X property[133X
  [33X[1;0Y[29X[2XIsoclinicStemDomain[102X( [3XG[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XAllStemGroupIds[102X( [3Xn[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XAllStemGroupFamilies[102X( [3Xn[103X ) [32X operation[133X
  
  [33X[0;0YA  group  [22XG[122X  is  a [13Xstem group[113X if [22XZ(G) ≤ [G,G][122X. Every group is isoclinic to a
  stem  group,  but distinct stem groups may be isoclinic. For example, groups
  [22XD_8, Q_8[122X are two isoclinic stem groups.[133X
  
  [33X[0;0YThe function [10XIsoclinicStemDomain [110X returns a stem group isoclinic to [22XG[122X.[133X
  
  [33X[0;0YThe  function [10XAllStemGroupIds[110X returns the [10XIdGroup[110X list of the stem groups of
  a   specified   size,  while  [10XAllStemGroupFamilies[110X  splits  this  list  into
  isoclinism classes.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XDerivedSubgroup(G);     [127X[104X
    [4X[28XGroup([ f3, f5 ])[128X[104X
    [4X[25Xgap>[125X [27XIsStemDomain( G );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsoclinicStemDomain( G );[127X[104X
    [4X[28X<pc group of size 16 with 4 generators>[128X[104X
    [4X[25Xgap>[125X [27XAllStemGroupIds( 32 );[127X[104X
    [4X[28X[ [ 32, 6 ], [ 32, 7 ], [ 32, 8 ], [ 32, 18 ], [ 32, 19 ], [ 32, 20 ], [128X[104X
    [4X[28X  [ 32, 27 ], [ 32, 28 ], [ 32, 29 ], [ 32, 30 ], [ 32, 31 ], [ 32, 32 ], [128X[104X
    [4X[28X  [ 32, 33 ], [ 32, 34 ], [ 32, 35 ], [ 32, 43 ], [ 32, 44 ], [ 32, 49 ], [128X[104X
    [4X[28X  [ 32, 50 ] ][128X[104X
    [4X[25Xgap>[125X [27XAllStemGroupFamilies( 32 );[127X[104X
    [4X[28X[ [ [ 32, 6 ], [ 32, 7 ], [ 32, 8 ] ], [ [ 32, 18 ], [ 32, 19 ], [ 32, 20 ] ],[128X[104X
    [4X[28X  [ [ 32, 27 ], [ 32, 28 ], [ 32, 29 ], [ 32, 30 ], [ 32, 31 ], [ 32, 32 ], [128X[104X
    [4X[28X      [ 32, 33 ], [ 32, 34 ], [ 32, 35 ] ], [ [ 32, 43 ], [ 32, 44 ] ], [128X[104X
    [4X[28X  [ [ 32, 49 ], [ 32, 50 ] ] ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X4.2-3 IsoclinicRank[101X
  
  [33X[1;0Y[29X[2XIsoclinicRank[102X( [3XG[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XIsoclinicMiddleLength[102X( [3XG[103X ) [32X attribute[133X
  
  [33X[0;0YLet [22XG[122X be a finite [22Xp[122X-group. Then [22Xlog_p |[G,G] / (Z(G) ∩ [G,G])|[122X is called the
  [13Xmiddle length[113X of [22XG[122X. Also [22Xlog_p |Z(G) ∩ [G,G]| + log_p |G/Z(G)|[122X is called the
  [13Xrank[113X  of  [22XG[122X. These invariants appear in the tables of isoclinism families of
  groups of order 128 in [JNO90].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XIsoclinicMiddleLength( G );[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27XIsoclinicRank( G );[127X[104X
    [4X[28X4[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X4.3 [33X[0;0YIsoclinism for crossed modules[133X[101X
  
  [1X4.3-1 Isoclinism[101X
  
  [33X[1;0Y[29X[2XIsoclinism[102X( [3XX0[103X, [3XY0[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XAreIsoclinicDomains[102X( [3XX0[103X, [3XY0[103X ) [32X operation[133X
  
  [33X[0;0YLet [22XcalX,calY[122X be crossed modules with central quotients [22XQ(calX)[122X and [22XQ(calY)[122X,
  and derived subcrossed modules [22X[calX,calX][122X and [22X[calY,calY][122X respectively. Let
  [22Xc_calX  : Q(calX) × Q(calX) -> [calX,calX][122X and [22Xc_calY : Q(calY) × Q(calY) ->
  [calY,calY][122X  be the two commutator maps. An [13Xisoclinism[113X [22XcalX ∼ calY[122X is a pair
  of  bijective  morphisms  [22X(η,ξ)[122X  where  [22Xη  :  Q(calX)  ->  Q(calY)[122X  and  [22Xξ :
  [calX,calX]  ->  [calY,calY][122X  such  that  [22Xc_calX  *  ξ  =  (η × η) * c_calY[122X.
  Isoclinism  is  an equivalence relation, and all abelian crossed modules are
  isoclinic to the trivial crossed module.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XC8 := Cat1Group( 16, 8, 2 );;[127X[104X
    [4X[25Xgap>[125X [27XX8 := XMod(C8);  IdGroup( X8 );[127X[104X
    [4X[28X[Group( [ f1*f2*f3, f3, f4 ] )->Group( [ f2, f2 ] )][128X[104X
    [4X[28X[ [ 8, 1 ], [ 2, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27XC9 := Cat1Group( 32, 9, 2 );[127X[104X
    [4X[28X[(C8 x C2) : C2 => Group( [ f2, f2 ] )][128X[104X
    [4X[25Xgap>[125X [27XX9 := XMod( C9 );  IdGroup( X9 );[127X[104X
    [4X[28X[Group( [ f1*f2*f3, f3, f4, f5 ] )->Group( [ f2, f2 ] )][128X[104X
    [4X[28X[ [ 16, 5 ], [ 2, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27XAreIsoclinicDomains( X8, X9 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xism89 := Isoclinism( X8, X9 );;[127X[104X
    [4X[25Xgap>[125X [27XDisplay( ism89 );[127X[104X
    [4X[28X[ [[Group( [ f1, f2, <identity> of ... ] ) -> Group( [ f2, f2 ] )] => [Group( [128X[104X
    [4X[28X    [ f1, f2, <identity> of ..., <identity> of ... ] ) -> Group( [128X[104X
    [4X[28X    [ f2, f2 ] )]], [128X[104X
    [4X[28X  [[Group( [ f3 ] ) -> Group( <identity> of ... )] => [Group( [128X[104X
    [4X[28X    [ f3 ] ) -> Group( <identity> of ... )]] ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X4.3-2 IsStemDomain[101X
  
  [33X[1;0Y[29X[2XIsStemDomain[102X( [3XX0[103X ) [32X property[133X
  [33X[1;0Y[29X[2XIsoclinicStemDomain[102X( [3XX0[103X ) [32X attribute[133X
  
  [33X[0;0YA  crossed  module  [22XcalX[122X  is a [13Xstem crossed module[113X if [22XZ(calX) ≤ [calX,calX][122X.
  Every  crossed  module  is  isoclinic to a stem crossed module, but distinct
  stem crossed modules may be isoclinic.[133X
  
  [33X[0;0YA method for [10XIsoclinicStemDomain[110X has yet to be implemented.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XIsStemDomain(X8);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsStemDomain(X9);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X4.3-3 IsoclinicRank[101X
  
  [33X[1;0Y[29X[2XIsoclinicRank[102X( [3XX0[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XIsoclinicMiddleLength[102X( [3XX0[103X ) [32X attribute[133X
  
  [33X[0;0YThe formulae in subsection [14X4.2-3[114X are applied to the crossed module.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XIsoclinicMiddleLength(X8);[127X[104X
    [4X[28X[ 1, 0 ][128X[104X
    [4X[25Xgap>[125X [27XIsoclinicRank(X8);        [127X[104X
    [4X[28X[ 3, 1 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
