  
  [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 FactorXMod[101X
  
  [29X[2XFactorXMod[102X( [3XX1[103X, [3XX2[103X ) [32X operation
  [29X[2XNaturalMorphismByNormalSubXMod[102X( [3XX1[103X, [3XX2[103X ) [32X operation
  
  [33X[0;0YWhen  [22XmathcalX_2  =  (∂_2  :  S_2  ->  R_2)[122X is a normal subcrossed module of
  [22XmathcalX_1  =  (∂_1  : S_1 -> R_1)[122X, then the quotient crossed module is [22X(∂ :
  S_2/S_1 -> R_2/R_1)[122X with the induced boundary and action maps.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xd24 := DihedralGroup(24);;  SetName( d24, "d24" );[127X[104X
    [4X[25Xgap>[125X [27XX24 := XModByAutomorphismGroup( d24 );;  Size(X24);[127X[104X
    [4X[28X[ 24, 48 ][128X[104X
    [4X[25Xgap>[125X [27Xnsx := NormalSubXMods( X24 );; [127X[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 [27XSize( Xn1 );[127X[104X
    [4X[28X[ 4, 8 ][128X[104X
    [4X[25Xgap>[125X [27Xnat1 := NaturalMorphismByNormalSubXMod( X24, Xn1 ); [127X[104X
    [4X[28X[[d24->PAut(d24)] => [..]][128X[104X
    [4X[25Xgap>[125X [27XQn1 := FactorXMod( X24, Xn1 );; [127X[104X
    [4X[25Xgap>[125X [27XSize( Qn1 );[127X[104X
    [4X[28X[ 6, 6 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X4.1-2 IntersectionSubXMods[101X
  
  [29X[2XIntersectionSubXMods[102X( [3XX0[103X, [3XX1[103X, [3XX2[103X ) [32X operation
  
  [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[28X[d24->Group( [ f1*f3, f2, f5 ] )][128X[104X
    [4X[25Xgap>[125X [27Xpos3 := Position( ids, [ [12,2], [24,5] ] );;[127X[104X
    [4X[25Xgap>[125X [27XXn3 := nsx[pos3]; [127X[104X
    [4X[28X[Group( [ f2, f3, f4 ] )->Group( [ f1, f2, f4, f5 ] )][128X[104X
    [4X[25Xgap>[125X [27XXn23 := IntersectionSubXMods( X24, Xn2, Xn3 );[127X[104X
    [4X[28X[Group( [ f2, f3, f4 ] )->Group( [ f2, f5, f2^2, f2*f5, f2^2*f5 ] )][128X[104X
    [4X[25Xgap>[125X [27X[ Size(Xn2), Size(Xn3), Size(Xn23) ];[127X[104X
    [4X[28X[ [ 24, 12 ], [ 12, 24 ], [ 12, 6 ] ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X4.1-3 Displacement[101X
  
  [29X[2XDisplacement[102X( [3Xalpha[103X, [3Xr[103X, [3Xs[103X ) [32X operation
  [29X[2XDisplacementSubgroup[102X( [3XX0[103X ) [32X attribute
  
  [33X[0;0YCommutators  may  be  written  [22X[r,q] = r^-1q^-1rq = (q^-1)^rq = r^-1r^q[122X, and
  satisfy identities[133X
  
  
        [33X[1;6Y[24X[33X[0;0Y[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[133X
  
  
  [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 [22XmathcalX = (∂ : S
  ->  R)[122X  is  a pre-crossed module, the first crossed module axiom requires [22X∂⟨
  r,s  ⟩ = [r,∂ s][122X. For a given action [22Xα[122X the [10XDisplacement[110X 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
  
  
        [33X[1;6Y[24X[33X[0;0Y\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[133X
  
  
  [33X[0;0YThe  [10XDisplacementSubgroup[110X  of  [22XmathcalX[122X  is the subgroup [22XDisp(mathcalX)[122X of [22XS[122X
  generated  by  these  displacements.  The  identities  imply  [22X⟨  r,s ⟩^t = ⟨
  r,st^r^-1} ⟩ ⟨ r^-1,t ⟩[122X, so [22XDisp(mathcalX)[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 [27XSn4 := Source(Xn4);; [127X[104X
    [4X[25Xgap>[125X [27XRn4 := Range(Xn4);; [127X[104X
    [4X[25Xgap>[125X [27Xr := Rn4.1;;  s := Sn4.1;; [127X[104X
    [4X[25Xgap>[125X [27Xd := Displacement( XModAction(Xn4), r, s );[127X[104X
    [4X[28Xf4[128X[104X
    [4X[25Xgap>[125X [27Xbn4 := Boundary( Xn4 );;[127X[104X
    [4X[25Xgap>[125X [27XImage( bn4, d ) = Comm( r, Image( bn4, s ) );  [127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XDisplacementSubgroup( Xn4 );[127X[104X
    [4X[28XGroup([ f4 ])[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X4.1-4 CommutatorSubXMod[101X
  
  [29X[2XCommutatorSubXMod[102X( [3XX[103X, [3XX1[103X, [3XX2[103X ) [32X operation
  [29X[2XCrossActionSubgroup[102X( [3XX[103X, [3XX1[103X, [3XX2[103X ) [32X operation
  
  [33X[0;0YWhen  [22XmathcalX_1 = (N -> Q), mathcalX_2 = (M -> P)[122X are two normal subcrossed
  modules  of  [22XmathcalX  = (∂ : 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[mathcalX_1,mathcalX_2][122X  has this subgroup as source and [22X[P,Q][122X as range, and
  is normal in [22XmathcalX[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XCrossActionSubgroup( X24, Xn2, Xn3 );[127X[104X
    [4X[28XGroup([ f2 ])[128X[104X
    [4X[25Xgap>[125X [27XCn23 := CommutatorSubXMod( X24, Xn2, Xn3 );[127X[104X
    [4X[28X[Group( [ f2 ] )->Group( [ f2, f5 ] )][128X[104X
    [4X[25Xgap>[125X [27XSize(Cn23);[127X[104X
    [4X[28X[ 12, 6 ][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
  
  [29X[2XDerivedSubXMod[102X( [3XX0[103X ) [32X attribute
  
  [33X[0;0YThe   [10XDerivedSubXMod[110X   of   [22XmathcalX[122X   is   the   normal  subcrossed  module
  [22X[mathcalX,mathcalX]  =  (∂'  :  Disp(mathcalX)  ->  [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[28X[Group( [ f4 ] )->Group( [ f2 ] )][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X4.1-6 FixedPointSubgroupXMod[101X
  
  [29X[2XFixedPointSubgroupXMod[102X( [3XX0[103X, [3XT[103X, [3XQ[103X ) [32X operation
  [29X[2XStabilizerSubgroupXMod[102X( [3XX0[103X, [3XT[103X, [3XQ[103X ) [32X operation
  
  [33X[0;0YThe  [10XFixedPointSubgroupXMod(X,T,Q)[110X for [22XmathcalX=(∂ : S -> R)[122X is the subgroup
  [22XFix(mathcalX,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   [22XmathcalX[122X   is   the   subgroup
  [22XStab(mathcalX,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([ f3*f4 ])[128X[104X
    [4X[25Xgap>[125X [27Xstab := StabilizerSubgroupXMod( Xn4, Sn4, Rn4 );[127X[104X
    [4X[28XGroup([ f5, f2*f3 ])[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X4.1-7 CentreXMod[101X
  
  [29X[2XCentreXMod[102X( [3XX0[103X ) [32X attribute
  [29X[2XCentralizer[102X( [3XX[103X, [3XY[103X ) [32X operation
  [29X[2XNormalizer[102X( [3XX[103X, [3XY[103X ) [32X operation
  
  [33X[0;0YThe  [13Xcentre[113X  [22XZ(mathcalX)[122X  of [22XmathcalX = (∂ : S -> R)[122X has as source the fixed
  point  subgroup  [22XFix(mathcalX,S,R)[122X.  The  range  is  the intersection of the
  centre [22XZ(R)[122X with the stabilizer subgroup.[133X
  
  [33X[0;0YWhen  [22XmathcalY = (T -> Q)[122X is a subcrossed module of [22XmathcalX = (∂ : S -> R)[122X,
  the  [13Xcentralizer[113X  [22XC_mathcalX}(mathcalY)[122X  of [22XmathcalY[122X has as source the fixed
  point  subgroup  [22XFix(mathcalX,S,Q)[122X.  The  range  is  the intersection of the
  centralizer [22XC_R(Q)[122X with [22XStab(mathcalX,T,R)[122X.[133X
  
  [33X[0;0YThe  [13Xnormalizer[113X [22XN_mathcalX}(mathcalY)[122X of [22XmathcalY[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
  
  [29X[2XCentralQuotient[102X( [3XG[103X ) [32X attribute
  
  [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
  [10XXModByCentralExtension[110X (see [14X2.1[114X).[133X
  
  [33X[0;0YSimilarly,  the central quotient of a crossed module [22XmathcalX[122X is the crossed
  square [22X(mathcalX ⇒ mathcalX/Z(mathcalX)[122X (see section [14X8.1[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XQ24 := CentralQuotient( d24 );  Size( Q24 );                     [127X[104X
    [4X[28X[d24->Group( [ f1, f2, f3 ] )][128X[104X
    [4X[28X[ 24, 12 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X4.1-9 IsAbelian2dGroup[101X
  
  [29X[2XIsAbelian2dGroup[102X( [3XX0[103X ) [32X property
  [29X[2XIsAspherical2dGroup[102X( [3XX0[103X ) [32X property
  [29X[2XIsSimplyConnected2dGroup[102X( [3XX0[103X ) [32X property
  [29X[2XIsFaithful2dGroup[102X( [3XX0[103X ) [32X property
  
  [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[ IsAbelian2dGroup(Xn4), IsAbelian2dGroup(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[ IsAspherical2dGroup(nsx[pos7]), IsAspherical2dGroup(X24) ];
[127X[104X
    [4X[28X[ true, false ]
[128X[104X
    [4X[25Xgap>[125X [27X[ IsSimplyConnected2dGroup(Xn4), IsSimplyConnected2dGroup(X24) ];[127X[104X
    [4X[28X[ true, true ][128X[104X
    [4X[25Xgap>[125X [27X[ IsFaithful2dGroup(Xn4), IsFaithful2dGroup(X24) ];              [127X[104X
    [4X[28X[ false, true ] [128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X4.1-10 LowerCentralSeriesOfXMod[101X
  
  [29X[2XLowerCentralSeriesOfXMod[102X( [3XX0[103X ) [32X attribute
  [29X[2XIsNilpotent2dGroup[102X( [3XX0[103X ) [32X property
  [29X[2XNilpotencyClass2dGroup[102X( [3XX0[103X ) [32X attribute
  
  [33X[0;0YLet  [22XmathcalY[122X be a subcrossed module of [22XmathcalX[122X. A [13X series of length[113X [22Xn[122X from
  [22XmathcalX[122X to [22XmathcalY[122X has the form[133X
  
  
        [33X[1;6Y[24X[33X[0;0Y\mathcal{X} ~=~ \mathcal{X}_0 ~\unrhd~ \mathcal{X}_1 ~\unrhd~ \cdots
        ~\unrhd~ \mathcal{X}_i ~\unrhd~ \cdots ~\unrhd~ \mathcal{X}_n ~=~
        \mathcal{Y} \quad (1 \leqslant i \leqslant n).[133X [124X[133X
  
  
  [33X[0;0YThe  quotients [22XmathcalF_i = mathcalX_i / mathcalX_i-1[122X are the [13Xfactors[113X of the
  series.[133X
  
  [33X[0;0YA  factor [22XmathcalF_i[122X is [13Xcentral[113X if [22XmathcalX_i-1 ⊴ mathcalX[122X and [22XmathcalF_i[122X is
  a subcrossed module of the centre of [22XmathcalX / mathcalX_i-1[122X.[133X
  
  [33X[0;0YA series is [13Xcentral[113X if all its factors are central.[133X
  
  [33X[0;0Y[22XmathcalX[122X is [13Xsoluble[113X if it has a series all of whose factors are abelian.[133X
  
  [33X[0;0Y[22XmathcalX[122X  is  [13Xnilpotent[113X  is it has a series all of whose factors are central
  factors of [22XmathcalX[122X.[133X
  
  [33X[0;0YThe [13Xlower central series[113X of [22XmathcalX[122X is the sequence (see [Nor87], p.77):[133X
  
  
        [33X[1;6Y[24X[33X[0;0Y\mathcal{X} ~=~ \Gamma_1(\mathcal{X}) ~\unrhd~ \Gamma_2(\mathcal{X})
        ~\unrhd~ \cdots \qquad \mbox{where} \qquad \Gamma_j(\mathcal{X}) ~=~ [
        \Gamma_{j-1}(\mathcal{X}), \mathcal{X}].[133X [124X[133X
  
  
  [33X[0;0YIf  [22XmathcalX[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(mathcalX)[122X is the trivial crossed module
  is the [13Xnilpotency class[113X of [22XmathcalX[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XLowerCentralSeries(X24);      [127X[104X
    [4X[28X[ [d24->PAut(d24)], [Group( [ f2 ] )->Group( [ f2, f5 ] )], [128X[104X
    [4X[28X  [Group( [ f3*f4^2 ] )->Group( [ f2 ] )], [Group( [ f4 ] )->Group( [ f2 ] )] [128X[104X
    [4X[28X ][128X[104X
    [4X[25Xgap>[125X [27XIsNilpotent2dGroup(X24);      [127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XNilpotencyClassOf2dGroup(X24);[127X[104X
    [4X[28X0[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X4.1-11 AllXMods[101X
  
  [29X[2XAllXMods[102X( [3Xargs[103X ) [32X function
  
  [33X[0;0YThe  global  function [10XAllXMods[110X may be called in three ways: as [10XAllXMods(S,R)[110X
  to  compute  all  crossed  modules  with  chosen source and range groups; as
  [10XAllXMods([n,m])[110X  to  compute  all  crossed  modules with a given size; or as
  [10XAllXMods(ord)[110X  to  compute  all crossed modules whose associated cat1-groups
  have a given size [10Xord[110X.[133X
  
  [33X[0;0YIn the example we see that there are [22X4[122X crossed modules [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 [27Xxc6s3 := AllXMods( SmallGroup(6,2), SmallGroup(6,1) );;   [127X[104X
    [4X[25Xgap>[125X [27XLength( xc6s3 );           [127X[104X
    [4X[28X4[128X[104X
    [4X[25Xgap>[125X [27Xx66 := AllXMods( [6,6] );;   [127X[104X
    [4X[25Xgap>[125X [27XLength( x66 );[127X[104X
    [4X[28X17[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
  
  [1X4.1-12 IsomorphismXMods[101X
  
  [29X[2XIsomorphismXMods[102X( [3XX1[103X, [3XX2[103X ) [32X operation
  [29X[2XAllXModsUpToIsomorphism[102X( [3Xlist[103X ) [32X operation
  
  [33X[0;0YThe  function  [10XIsomorphismXMods[110X  computes  an  isomorphism [22Xμ : mathcalX_1 ->
  mathcalX_2[122X,  provided one exists, or else returns [10Xfail[110X. In the example below
  we  see  that  the [22X17[122X crossed modules of size [10X[6,6][110X in [10Xx66[110X (see the previous
  subsection) fall into [22X9[122X isomorphism classes.[133X
  
  [33X[0;0YThe  function  [10XAllXModsUpToIsomorphism[110X  takes  a list of crossed modules and
  partitions them into isomorphism classes.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XIsomorphismXMods( x66[1], x66[2] );[127X[104X
    [4X[28X[[Group( [ f1, f2 ] )->Group( [ f1, f2 ] )] => [Group( [ f1, f2 ] )->Group( [128X[104X
    [4X[28X[ f1, f2 ] )]][128X[104X
    [4X[25Xgap>[125X [27Xiso66 := AllXModsUpToIsomorphism( x66 );;  Length( iso66 ); [127X[104X
    [4X[28X9 [128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X4.2 [33X[0;0YIsoclinism for groups[133X[101X
  
  [1X4.2-1 Isoclinism[101X
  
  [29X[2XIsoclinism[102X( [3XG[103X, [3XH[103X ) [32X operation
  [29X[2XAreIsoclinicDomains[102X( [3XG[103X, [3XH[103X ) [32X operation
  
  [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
  
  [29X[2XIsStemDomain[102X( [3XG[103X ) [32X property
  [29X[2XIsoclinicStemDomain[102X( [3XG[103X ) [32X attribute
  [29X[2XAllStemGroupIds[102X( [3Xn[103X ) [32X operation
  [29X[2XAllStemGroupFamilies[102X( [3Xn[103X ) [32X operation
  
  [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
  
  [29X[2XIsoclinicRank[102X( [3XG[103X ) [32X attribute
  [29X[2XIsoclinicMiddleLength[102X( [3XG[103X ) [32X attribute
  
  [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
  
  [29X[2XIsoclinism[102X( [3XX0[103X, [3XY0[103X ) [32X operation
  [29X[2XAreIsoclinicDomains[102X( [3XX0[103X, [3XY0[103X ) [32X operation
  
  [33X[0;0YLet  [22XmathcalX,mathcalY[122X be crossed modules with central quotients [22XQ(mathcalX)[122X
  and  [22XQ(mathcalY)[122X,  and  derived  subcrossed  modules [22X[mathcalX,mathcalX][122X and
  [22X[mathcalY,mathcalY][122X respectively. Let [22Xc_mathcalX : Q(mathcalX) × Q(mathcalX)
  ->  [mathcalX,mathcalX][122X  and  [22Xc_mathcalY  :  Q(mathcalY)  ×  Q(mathcalY)  ->
  [mathcalY,mathcalY][122X  be  the  two  commutator maps. An [13Xisoclinism[113X [22XmathcalX ∼
  mathcalY[122X  is  a  pair  of bijective morphisms [22X(η,ξ)[122X where [22Xη : Q(mathcalX) ->
  Q(mathcalY)[122X  and  [22Xξ  :  [mathcalX,mathcalX] -> [mathcalY,mathcalY][122X such that
  [22Xc_mathcalX  *  ξ  =  (η  ×  η)  *  c_mathcalY[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 := Cat1(16,8,1);;[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 := Cat1(32,9,1);[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 ] )->Group( [ f2 ] )] => [Group( [ f1 ] )->Group( [ 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
  
  [29X[2XIsStemDomain[102X( [3XX0[103X ) [32X property
  [29X[2XIsoclinicStemDomain[102X( [3XX0[103X ) [32X property
  
  [33X[0;0YA  crossed  module  [22XmathcalX[122X  is  a  [13Xstem  crossed  module[113X  if [22XZ(mathcalX) ≤
  [mathcalX,mathcalX][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
  
  [29X[2XIsoclinicRank[102X( [3XX0[103X ) [32X attribute
  [29X[2XIsoclinicMiddleLength[102X( [3XX0[103X ) [32X attribute
  
  [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
  
