Isoclinism of groups and crossed modules This chapter describes some functions written by Alper Odabaş and Enver Uslu, and reported in their paper . Section contains some additional basic functions for crossed modules, constructing quotients, centres, centralizers and normalizers. In Sections and 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, .
More operations for crossed modules When \mathcal{X}_2 = (\partial_2 : S_2 \to R_2) is a normal sub-precrossed module of \mathcal{X}_1 = (\partial_1 : S_1 \to R_1), then the quotient precrossed module is (\partial : S_2/S_1 \to R_2/R_1) with the induced boundary and action maps. Quotienting a precrossed module by it's Peiffer subgroup is a special case of this construction.

gap> d24 := DihedralGroup( IsPermGroup, 24 );; gap> SetName( d24, "d24" ); gap> Y24 := XModByAutomorphismGroup( d24 );; gap> Size( Y24 ); [ 24, 48 ] gap> X24 := Image( IsomorphismPerm2DimensionalGroup( Y24 ) ); [d24->Group( [ ( 2, 4), ( 1, 2, 3, 4)( 5, 8)( 6, 9)( 7,10), ( 6,10)( 7, 9), ( 5, 9, 7)( 6,10, 8) ] )] gap> nsx := NormalSubXMods( X24 );; gap> Length( nsx ); 40 gap> ids := List( nsx, n -> IdGroup(n) );; gap> pos1 := Position( ids, [ [4,1], [8,3] ] );; gap> Xn1 := nsx[pos1]; [Group( [ f2*f4^2, f3*f4 ] )->Group( [ f3, f4, f5 ] )] gap> nat1 := NaturalMorphismByNormalSubPreXMod( X24, Xn1 );; gap> Qn1 := FactorPreXMod( X24, Xn1 );; gap> [ Size( Xn1 ), Size( Qn1 ) ]; [ [ 4, 8 ], [ 6, 6 ] ]
When X1,X2 are subcrossed modules of X0, then the source and range of their intersection are the intersections of the sources and ranges of X1 and X2 respectively.

gap> pos2 := Position( ids, [ [24,6], [12,4] ] );; gap> Xn2 := nsx[pos2];; gap> IdGroup( Xn2 ); [ [ 24, 6 ], [ 12, 4 ] ] gap> pos3 := Position( ids, [ [12,2], [24,5] ] );; gap> Xn3 := nsx[pos3];; gap> IdGroup( Xn3 ); [ [ 12, 2 ], [ 24, 5 ] ] gap> Xn23 := IntersectionSubXMods( X24, Xn2, Xn3 );; gap> IdGroup( Xn23 ); [ [ 12, 2 ], [ 6, 2 ] ]
Commutators may be written [r,q] = r^{-1}q^{-1}rq = (q^{-1})^rq = r^{-1}r^q, and satisfy identities [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]. In a similar way, when a group R acts on a group S, the displacement of s \in S by r \in R is defined to be \langle r,s \rangle := (s^{-1})^rs \in S. When \mathcal{X} = (\partial : S \to R) is a pre-crossed module, the first crossed module axiom requires \partial\langle r,s \rangle = [r,\partial s]. When \alpha is the action of \mathcal{X}, the function may be used to calculate \langle r,s \rangle. Displacements satisfy the following identities, where s,t \in S,~ p,q,r \in R: \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. The operation applied to X0,Q,T is the subgroup of S consisting of all the displacements \langle r,s \rangle, r \in Q \leqslant R, s \in T \leqslant S. The of \mathcal{X} is the subgroup \Disp(\mathcal{X}) of S given by DisplacementGroup(X0,R,S). The identities imply \langle r,s \rangle^t = \langle r,st^{r^{-1}} \rangle \langle r^{-1},t \rangle, so \Disp(\mathcal{X}) is normal in S.

gap> pos4 := Position( ids, [ [6,2], [24,14] ] );; gap> Xn4 := nsx[pos4];; gap> bn4 := Boundary( Xn4 );; gap> Sn4 := Source(Xn4);; gap> Rn4 := Range(Xn4);; gap> genRn4 := GeneratorsOfGroup( Rn4 );; gap> L := List( genRn4, g -> ( Order(g) = 2 ) and > not ( IsNormal( Rn4, Subgroup( Rn4, [g] ) ) ) );; gap> pos := Position( L, true );; gap> s := Sn4.1; r := genRn4[pos]; (1,3,5,7,9,11)(2,4,6,8,10,12) (6,10)(7,9) gap> act := XModAction( Xn4 );; gap> d := Displacement( act, r, s ); (1,5,9)(2,6,10)(3,7,11)(4,8,12) gap> Image( bn4, d ) = Comm( r, Image( bn4, s ) ); true gap> Qn4 := Subgroup( Rn4, [ (6,10)(7,9), (1,3), (2,4) ] );; gap> Tn4 := Subgroup( Sn4, [ (1,3,5,7,9,11)(2,4,6,8,10,12) ] );; gap> DisplacementGroup( Xn4, Qn4, Tn4 ); Group([ (1,5,9)(2,6,10)(3,7,11)(4,8,12) ]) gap> DisplacementSubgroup( Xn4 ); Group([ (1,5,9)(2,6,10)(3,7,11)(4,8,12) ])
When \mathcal{X}_1 = (N \to Q), \mathcal{X}_2 = (M \to P) are two normal subcrossed modules of \mathcal{X} = (\partial : S \to R), the displacements \langle p,n \rangle and \langle q,m \rangle all map by \partial into [Q,P]. These displacements form a normal subgroup of S, called the CrossActionSubgroup. The CommutatorSubXMod [\mathcal{X}_1,\mathcal{X}_2] has this subgroup as source and [P,Q] as range, and is normal in \mathcal{X}.

gap> CAn23 := CrossActionSubgroup( X24, Xn2, Xn3 );; gap> IdGroup( CAn23 ); [ 12, 2 ] gap> Cn23 := CommutatorSubXMod( X24, Xn2, Xn3 );; gap> IdGroup( Cn23 ); [ [ 12, 2 ], [ 6, 2 ] ] gap> Xn23 = Cn23; true
The DerivedSubXMod of \mathcal{X} is the normal subcrossed module [\mathcal{X},\mathcal{X}] = (\partial' : \Disp(\mathcal{X}) \to [R,R]) where \partial' is the restriction of \partial (see page 66 of Norrie's thesis ).

gap> DXn4 := DerivedSubXMod( Xn4 );; gap> IdGroup( DXn4 ); [ [ 3, 1 ], [ 3, 1 ] ]
The FixedPointSubgroupXMod(X,T,Q) for \mathcal{X}=(\partial : S \to R) is the subgroup \Fix(\mathcal{X},T,Q) of elements t \in T \leqslant S individually fixed under the action of Q \leqslant R.

The StabilizerSubgroupXMod(X,T,Q) for \mathcal{X} is the subgroup \Stab(\mathcal{X},T,Q) of Q \leqslant R whose elements act trivially on the whole of T \leqslant S (see page 19 of Norrie's thesis ).

gap> fix := FixedPointSubgroupXMod( Xn4, Sn4, Rn4 ); Group([ (1,7)(2,8)(3,9)(4,10)(5,11)(6,12) ]) gap> stab := StabilizerSubgroupXMod( Xn4, Sn4, Rn4 );; gap> IdGroup( stab ); [ 12, 5 ]
The centre Z(\mathcal{X}) of \mathcal{X} = (\partial : S \to R) has as source the fixed point subgroup \Fix(\mathcal{X},S,R). The range is the intersection of the centre Z(R) with the stabilizer subgroup.

When \mathcal{Y} = (T \to Q) is a subcrossed module of \mathcal{X} = (\partial : S \to R), the centralizer C_{\mathcal{X}}(\mathcal{Y}) of \mathcal{Y} has as source the fixed point subgroup \Fix(\mathcal{X},S,Q). The range is the intersection of the centralizer C_R(Q) with \Stab(\mathcal{X},T,R).

The normalizer N_{\mathcal{X}}(\mathcal{Y}) of \mathcal{Y} has as source the subgroup of S consisting of the displacements \langle s,q \rangle which lie in S.

gap> ZXn4 := CentreXMod( Xn4 ); [Group( [ f3*f4 ] )->Group( [ f3, f5 ] )] gap> IdGroup( ZXn4 ); [ [ 2, 1 ], [ 4, 2 ] ] gap> CDXn4 := Centralizer( Xn4, DXn4 ); [Group( [ f3*f4 ] )->Group( [ f2 ] )] gap> IdGroup( CDXn4 ); [ [ 2, 1 ], [ 3, 1 ] ] gap> NDXn4 := Normalizer( Xn4, DXn4 ); [Group( <identity> of ... )->Group( [ f5, f2*f3 ] )] gap> IdGroup( NDXn4 ); [ [ 1, 1 ], [ 12, 5 ] ]
The CentralQuotient of a group G is the crossed module (G \to G/Z(G)) with the natural homomorphism as the boundary map. This is a special case of .

Similarly, the central quotient of a crossed module \mathcal{X} is the crossed square (\mathcal{X} \Rightarrow \mathcal{X}/Z(\mathcal{X})) (see section ).

gap> Q24 := CentralQuotient( d24); IdGroup( Q24 ); [d24->d24/Z(d24)] [ [ 24, 6 ], [ 12, 4 ] ]
A crossed module is abelian if it equal to its centre. This is the case when the range group is abelian and the action is trivial.

A crossed module is aspherical if the boundary has trivial kernel.

A crossed module is simply connected if the boundary has trivial cokernel.

A crossed module is faithful if the action is faithful.

gap> [ IsAbelian2DimensionalGroup(Xn4), IsAbelian2DimensionalGroup(X24) ]; [ false, false ] gap> pos7 := Position( ids, [ [3,1], [6,1] ] );; gap> [ IsAspherical2DimensionalGroup(nsx[pos7]), IsAspherical2DimensionalGroup(X24) ]; [ true, false ] gap> [ IsSimplyConnected2DimensionalGroup(Xn4), IsSimplyConnected2DimensionalGroup(X24) ]; [ true, true ] gap> [ IsFaithful2DimensionalGroup(Xn4), IsFaithful2DimensionalGroup(X24) ]; [ false, true ]
Let \mathcal{Y} be a subcrossed module of \mathcal{X}. A series of length n from \mathcal{X} to \mathcal{Y} has the form \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). The quotients \mathcal{F}_i = \mathcal{X}_i / \mathcal{X}_{i-1} are the factors of the series.

A factor \mathcal{F}_i is central if \mathcal{X}_{i-1} \unlhd \mathcal{X} and \mathcal{F}_i is a subcrossed module of the centre of \mathcal{X} / \mathcal{X}_{i-1}.

A series is central if all its factors are central.

\mathcal{X} is soluble if it has a series all of whose factors are abelian.

\mathcal{X} is nilpotent is it has a series all of whose factors are central factors of \mathcal{X}.

The lower central series of \mathcal{X} is the sequence (see , p.77): \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}]. If \mathcal{X} is nilpotent, then its lower central series is its most rapidly descending central series.

The least integer c such that \Gamma_{c+1}(\mathcal{X}) is the trivial crossed module is the nilpotency class of \mathcal{X}.

gap> lcs := LowerCentralSeries( X24 );; gap> List( lcs, g -> IdGroup(g) ); [ [ [ 24, 6 ], [ 48, 38 ] ], [ [ 12, 2 ], [ 6, 2 ] ], [ [ 6, 2 ], [ 3, 1 ] ], [ [ 3, 1 ], [ 3, 1 ] ] ] gap> IsNilpotent2DimensionalGroup( X24 ); false gap> NilpotencyClassOf2DimensionalGroup( X24 ); 0
The global function AllXMods may be called in three ways: as AllXMods(S,R) to compute all crossed modules with chosen source and range groups; as AllXMods([n,m]) to compute all crossed modules with a given size; or as AllXMods(ord) to compute all crossed modules whose associated cat1-groups have a given size ord.

In the example we see that there are 4 crossed modules (C_6 \to S_3); forming a subset of the 17 crossed modules with size [6,6]; and that these form a subset of the 205 crossed modules whose cat1-group has size 36. There are 40 precrossed modules with size [6,6].

gap> xc6s3 := AllXMods( SmallGroup(6,2), SmallGroup(6,1) );; gap> Length( xc6s3 ); 4 gap> x66 := AllXMods( [6,6] );; gap> Length( x66 ); 17 gap> x36 := AllXMods( 36 );; gap> Length( x36 ); 205 gap> size36 := List( x36, x -> [ Size(Source(x)), Size(Range(x)) ] );; gap> Collected( size36 ); [ [ [ 1, 36 ], 14 ], [ [ 2, 18 ], 7 ], [ [ 3, 12 ], 21 ], [ [ 4, 9 ], 14 ], [ [ 6, 6 ], 17 ], [ [ 9, 4 ], 102 ], [ [ 12, 3 ], 8 ], [ [ 18, 2 ], 18 ], [ [ 36, 1 ], 4 ] ]
The function IsomorphismXMods computes an isomorphism \mu : \mathcal{X}_1 \to \mathcal{X}_2, provided one exists, or else returns fail. In the example below we see that the 17 crossed modules of size [6,6] in x66 (see the previous subsection) fall into 9 isomorphism classes.

The function AllXModsUpToIsomorphism takes a list of crossed modules and partitions them into isomorphism classes.

gap> IsomorphismXMods( x66[1], x66[2] ); [[Group( [ f1, f2 ] )->Group( [ f1, f2 ] )] => [Group( [ f1, f2 ] )->Group( [ f1, f2 ] )]] gap> iso66 := AllXModsUpToIsomorphism( x66 );; Length( iso66 ); 9
Isoclinism for groups Let G,H be groups with central quotients Q(G) and Q(H) and derived subgroups [G,G] and [H,H] respectively. Let c_G : G/Z(G) \times G/Z(G) \to [G,G] and c_H : H/Z(H) \times H/Z(H) \to [H,H] be the two commutator maps. An isoclinism G \sim H is a pair of isomorphisms (\eta,\xi) where \eta : G/Z(G) \to H/Z(H) and \xi : [G,G] \to [H,H] such that c_G * \xi = (\eta \times \eta) * c_H. Isoclinism is an equivalence relation, and all abelian groups are isoclinic to the trivial group.

gap> G := SmallGroup( 64, 6 );; StructureDescription( G ); "(C8 x C4) : C2" gap> QG := CentralQuotient( G );; IdGroup( QG ); [ [ 64, 6 ], [ 8, 3 ] ] gap> H := SmallGroup( 32, 41 );; StructureDescription( H ); "C2 x Q16" gap> QH := CentralQuotient( H );; IdGroup( QH ); [ [ 32, 41 ], [ 8, 3 ] ] gap> Isoclinism( G, H ); [ [ f1, f2, f3 ] -> [ f1, f2*f3, f3 ], [ f3, f5 ] -> [ f4*f5, f5 ] ] gap> K := SmallGroup( 32, 43 );; StructureDescription( K ); "(C2 x D8) : C2" gap> QK := CentralQuotient( K );; IdGroup( QK ); [ [ 32, 43 ], [ 16, 11 ] ] gap> AreIsoclinicDomains( G, K ); false
A group G is a stem group if Z(G) \leq [G,G]. Every group is isoclinic to a stem group, but distinct stem groups may be isoclinic. For example, groups D_8, Q_8 are two isoclinic stem groups.

The function IsoclinicStemDomain returns a stem group isoclinic to G.

The function AllStemGroupIds returns the IdGroup list of the stem groups of a specified size, while AllStemGroupFamilies splits this list into isoclinism classes.

gap> DerivedSubgroup(G); Group([ f3, f5 ]) gap> IsStemDomain( G ); false gap> IsoclinicStemDomain( G ); <pc group of size 16 with 4 generators> gap> AllStemGroupIds( 32 ); [ [ 32, 6 ], [ 32, 7 ], [ 32, 8 ], [ 32, 18 ], [ 32, 19 ], [ 32, 20 ], [ 32, 27 ], [ 32, 28 ], [ 32, 29 ], [ 32, 30 ], [ 32, 31 ], [ 32, 32 ], [ 32, 33 ], [ 32, 34 ], [ 32, 35 ], [ 32, 43 ], [ 32, 44 ], [ 32, 49 ], [ 32, 50 ] ] gap> AllStemGroupFamilies( 32 ); [ [ [ 32, 6 ], [ 32, 7 ], [ 32, 8 ] ], [ [ 32, 18 ], [ 32, 19 ], [ 32, 20 ] ], [ [ 32, 27 ], [ 32, 28 ], [ 32, 29 ], [ 32, 30 ], [ 32, 31 ], [ 32, 32 ], [ 32, 33 ], [ 32, 34 ], [ 32, 35 ] ], [ [ 32, 43 ], [ 32, 44 ] ], [ [ 32, 49 ], [ 32, 50 ] ] ]
Let G be a finite p-group. Then \log_p |[G,G] / (Z(G) \cap [G,G])| is called the middle length of G. Also \log_p |Z(G) \cap [G,G]| + \log_p |G/Z(G)| is called the rank of G. These invariants appear in the tables of isoclinism families of groups of order 128 in .

gap> IsoclinicMiddleLength( G ); 1 gap> IsoclinicRank( G ); 4
Isoclinism for crossed modules Let \mathcal{X},\mathcal{Y} be crossed modules with central quotients Q(\mathcal{X}) and Q(\mathcal{Y}), and derived subcrossed modules [\mathcal{X},\mathcal{X}] and [\mathcal{Y},\mathcal{Y}] respectively. Let c_\mathcal{X} : Q(\mathcal{X}) \times Q(\mathcal{X}) \to [\mathcal{X},\mathcal{X}] and c_\mathcal{Y} : Q(\mathcal{Y}) \times Q(\mathcal{Y}) \to [\mathcal{Y},\mathcal{Y}] be the two commutator maps. An isoclinism \mathcal{X} \sim \mathcal{Y} is a pair of bijective morphisms (\eta,\xi) where \eta : Q(\mathcal{X}) \to Q(\mathcal{Y}) and \xi : [\mathcal{X},\mathcal{X}] \to [\mathcal{Y},\mathcal{Y}] such that c_\mathcal{X} * \xi = (\eta \times \eta) * c_\mathcal{Y}. Isoclinism is an equivalence relation, and all abelian crossed modules are isoclinic to the trivial crossed module.

gap> C8 := Cat1Group( 16, 8, 1 );; gap> X8 := XMod(C8); IdGroup( X8 ); [Group( [ f1*f2*f3, f3, f4 ] )->Group( [ f2, f2 ] )] [ [ 8, 1 ], [ 2, 1 ] ] gap> C9 := Cat1Group( 32, 9, 1 ); [(C8 x C2) : C2=>Group( [ f2, f2 ] )] gap> X9 := XMod( C9 ); IdGroup( X9 ); [Group( [ f1*f2*f3, f3, f4, f5 ] )->Group( [ f2, f2 ] )] [ [ 16, 5 ], [ 2, 1 ] ] gap> AreIsoclinicDomains( X8, X9 ); true gap> ism89 := Isoclinism( X8, X9 );; gap> Display( ism89 ); [ [[Group( [ f1, f2, <identity> of ... ] ) -> Group( [ f2, f2 ] )] => [Group( [ f1, f2, <identity> of ..., <identity> of ... ] ) -> Group( [ f2, f2 ] )]], [[Group( [ f3 ] ) -> Group( <identity> of ... )] => [Group( [ f3 ] ) -> Group( <identity> of ... )]] ]
A crossed module \mathcal{X} is a stem crossed module if Z(\mathcal{X}) \leq [\mathcal{X},\mathcal{X}]. Every crossed module is isoclinic to a stem crossed module, but distinct stem crossed modules may be isoclinic.

A method for IsoclinicStemDomain has yet to be implemented.

gap> IsStemDomain(X8); true gap> IsStemDomain(X9); false
The formulae in subsection are applied to the crossed module.

gap> IsoclinicMiddleLength(X8); [ 1, 0 ] gap> IsoclinicRank(X8); [ 3, 1 ]