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### 4 Isoclinism of groups and crossed modules

This chapter describes some functions written by Alper Odabaş and Enver Uslu, and reported in their paper [IOU16]. Section 4.1 contains some additional basic functions for crossed modules, constructing quotients, centres, centralizers and normalizers. In Sections 4.2 and 4.3 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].

#### 4.1 More operations for crossed modules

##### 4.1-1 FactorPreXMod
 ‣ FactorPreXMod( X1, X2 ) ( operation )
 ‣ NaturalMorphismByNormalSubPreXMod( X1, X2 ) ( operation )

When mathcalX_2 = (∂_2 : S_2 -> R_2) is a normal sub-precrossed module of mathcalX_1 = (∂_1 : S_1 -> R_1), then the quotient precrossed module is (∂ : S_2/S_1 -> 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(24);;  SetName( d24, "d24" );
gap> X24 := XModByAutomorphismGroup( d24 );;  Size(X24);
[ 24, 48 ]
gap> nsx := NormalSubXMods( X24 );;
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 );
[[d24->PAut(d24)] => [..]]
gap> Qn1 := FactorPreXMod( X24, Xn1 );;
gap> [ Size(Xn1), Size(Qn1) ];
[ [ 4, 8 ], [ 6, 6 ] ]



##### 4.1-2 IntersectionSubXMods
 ‣ IntersectionSubXMods( X0, X1, X2 ) ( operation )

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];
[d24->Group( [ f1*f3, f2, f5 ] )]
gap> pos3 := Position( ids, [ [12,2], [24,5] ] );;
gap> Xn3 := nsx[pos3];
[Group( [ f2, f3, f4 ] )->Group( [ f1, f2, f4, f5 ] )]
gap> Xn23 := IntersectionSubXMods( X24, Xn2, Xn3 );
[Group( [ f2, f3, f4 ] )->Group( [ f2, f5, f2^2, f2*f5, f2^2*f5 ] )]
gap> [ Size(Xn2), Size(Xn3), Size(Xn23) ];
[ [ 24, 12 ], [ 12, 24 ], [ 12, 6 ] ]



##### 4.1-3 Displacement
 ‣ Displacement( alpha, r, s ) ( operation )
 ‣ DisplacementSubgroup( X0 ) ( attribute )

Commutators may be written [r,q] = r^-1q^-1rq = (q^-1)^rq = r^-1r^q, and satisfy identities

In a similar way, when a group R acts on a group S, the displacement of s ∈ S by r ∈ R is defined to be ⟨ r,s ⟩ := (s^-1)^rs ∈ S. When mathcalX = (∂ : S -> R) is a pre-crossed module, the first crossed module axiom requires ∂⟨ r,s ⟩ = [r,∂ s]. For a given action α the Displacement function may be used to calculate ⟨ r,s ⟩. Displacements satisfy the following identities, where s,t ∈ S,~ p,q,r ∈ 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 DisplacementSubgroup of mathcalX is the subgroup Disp(mathcalX) of S generated by these displacements. The identities imply ⟨ r,s ⟩^t = ⟨ r,st^r^-1} ⟩ ⟨ r^-1,t ⟩, so Disp(mathcalX) is normal in S.


gap> pos4 := Position( ids, [ [6,2], [24,14] ] );;
gap> Xn4 := nsx[pos4];;
gap> Sn4 := Source(Xn4);;
gap> Rn4 := Range(Xn4);;
gap> r := Rn4.1;;  s := Sn4.1;;
gap> d := Displacement( XModAction(Xn4), r, s );
f4
gap> bn4 := Boundary( Xn4 );;
gap> Image( bn4, d ) = Comm( r, Image( bn4, s ) );
true
gap> DisplacementSubgroup( Xn4 );
Group([ f4 ])



##### 4.1-4 CommutatorSubXMod
 ‣ CommutatorSubXMod( X, X1, X2 ) ( operation )
 ‣ CrossActionSubgroup( X, X1, X2 ) ( operation )

When mathcalX_1 = (N -> Q), mathcalX_2 = (M -> P) are two normal subcrossed modules of mathcalX = (∂ : S -> R), the displacements ⟨ p,n ⟩ and ⟨ q,m ⟩ all map by into [Q,P]. These displacements form a normal subgroup of S, called the CrossActionSubgroup. The CommutatorSubXMod [mathcalX_1,mathcalX_2] has this subgroup as source and [P,Q] as range, and is normal in mathcalX.


gap> CrossActionSubgroup( X24, Xn2, Xn3 );
Group([ f2 ])
gap> Cn23 := CommutatorSubXMod( X24, Xn2, Xn3 );
[Group( [ f2 ] )->Group( [ f2, f5 ] )]
gap> Size(Cn23);
[ 12, 6 ]
gap> Xn23 = Cn23;
true



##### 4.1-5 DerivedSubXMod
 ‣ DerivedSubXMod( X0 ) ( attribute )

The DerivedSubXMod of mathcalX is the normal subcrossed module [mathcalX,mathcalX] = (∂' : Disp(mathcalX) -> [R,R]) where ∂' is the restriction of (see page 66 of Norrie's thesis [Nor87]).


gap> DXn4 := DerivedSubXMod( Xn4 );
[Group( [ f4 ] )->Group( [ f2 ] )]



##### 4.1-6 FixedPointSubgroupXMod
 ‣ FixedPointSubgroupXMod( X0, T, Q ) ( operation )
 ‣ StabilizerSubgroupXMod( X0, T, Q ) ( operation )

The FixedPointSubgroupXMod(X,T,Q) for mathcalX=(∂ : S -> R) is the subgroup Fix(mathcalX,T,Q) of elements t ∈ T leqslant S individually fixed under the action of Q leqslant R.

The StabilizerSubgroupXMod(X,T,Q) for mathcalX is the subgroup Stab(mathcalX,T,Q) of Q leqslant R whose elements act trivially on the whole of T leqslant S (see page 19 of Norrie's thesis [Nor87]).


gap> fix := FixedPointSubgroupXMod( Xn4, Sn4, Rn4 );
Group([ f3*f4 ])
gap> stab := StabilizerSubgroupXMod( Xn4, Sn4, Rn4 );
Group([ f5, f2*f3 ])



##### 4.1-7 CentreXMod
 ‣ CentreXMod( X0 ) ( attribute )
 ‣ Centralizer( X, Y ) ( operation )
 ‣ Normalizer( X, Y ) ( operation )

The centre Z(mathcalX) of mathcalX = (∂ : S -> R) has as source the fixed point subgroup Fix(mathcalX,S,R). The range is the intersection of the centre Z(R) with the stabilizer subgroup.

When mathcalY = (T -> Q) is a subcrossed module of mathcalX = (∂ : S -> R), the centralizer C_mathcalX}(mathcalY) of mathcalY has as source the fixed point subgroup Fix(mathcalX,S,Q). The range is the intersection of the centralizer C_R(Q) with Stab(mathcalX,T,R).

The normalizer N_mathcalX}(mathcalY) of mathcalY has as source the subgroup of S consisting of the displacements ⟨ s,q ⟩ 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 ] ]



##### 4.1-8 CentralQuotient
 ‣ CentralQuotient( G ) ( attribute )

The CentralQuotient of a group G is the crossed module (G -> G/Z(G)) with the natural homomorphism as the boundary map. This is a special case of XModByCentralExtension (see 2.1).

Similarly, the central quotient of a crossed module mathcalX is the crossed square (mathcalX ⇒ mathcalX/Z(mathcalX) (see section 8.2).


gap> Q24 := CentralQuotient( d24 );  Size( Q24 );
[d24->Group( [ f1, f2, f3 ] )]
[ 24, 12 ]



##### 4.1-9 IsAbelian2DimensionalGroup
 ‣ IsAbelian2DimensionalGroup( X0 ) ( property )
 ‣ IsAspherical2DimensionalGroup( X0 ) ( property )
 ‣ IsSimplyConnected2DimensionalGroup( X0 ) ( property )
 ‣ IsFaithful2DimensionalGroup( X0 ) ( property )

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 ]



##### 4.1-10 LowerCentralSeriesOfXMod
 ‣ LowerCentralSeriesOfXMod( X0 ) ( attribute )
 ‣ IsNilpotent2DimensionalGroup( X0 ) ( property )
 ‣ NilpotencyClass2DimensionalGroup( X0 ) ( attribute )

Let mathcalY be a subcrossed module of mathcalX. A series of length n from mathcalX to mathcalY 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 mathcalF_i = mathcalX_i / mathcalX_i-1 are the factors of the series.

A factor mathcalF_i is central if mathcalX_i-1 ⊴ mathcalX and mathcalF_i is a subcrossed module of the centre of mathcalX / mathcalX_i-1.

A series is central if all its factors are central.

mathcalX is soluble if it has a series all of whose factors are abelian.

mathcalX is nilpotent is it has a series all of whose factors are central factors of mathcalX.

The lower central series of mathcalX is the sequence (see [Nor87], 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 mathcalX is nilpotent, then its lower central series is its most rapidly descending central series.

The least integer c such that Γ_c+1(mathcalX) is the trivial crossed module is the nilpotency class of mathcalX.


gap> LowerCentralSeries(X24);
[ [d24->PAut(d24)], [Group( [ f2 ] )->Group( [ f2, f5 ] )],
[Group( [ f3*f4^2 ] )->Group( [ f2 ] )], [Group( [ f4 ] )->Group( [ f2 ] )]
]
gap> IsNilpotent2DimensionalGroup(X24);
false
gap> NilpotencyClassOf2DimensionalGroup(X24);
0



##### 4.1-11 AllXMods
 ‣ AllXMods( args ) ( function )

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 -> 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 ] ]



##### 4.1-12 IsomorphismXMods
 ‣ IsomorphismXMods( X1, X2 ) ( operation )
 ‣ AllXModsUpToIsomorphism( list ) ( operation )

The function IsomorphismXMods computes an isomorphism μ : mathcalX_1 -> mathcalX_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



#### 4.2 Isoclinism for groups

##### 4.2-1 Isoclinism
 ‣ Isoclinism( G, H ) ( operation )
 ‣ AreIsoclinicDomains( G, H ) ( operation )

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) × G/Z(G) -> [G,G] and c_H : H/Z(H) × H/Z(H) -> [H,H] be the two commutator maps. An isoclinism G ∼ H is a pair of isomorphisms (η,ξ) where η : G/Z(G) -> H/Z(H) and ξ : [G,G] -> [H,H] such that c_G * ξ = (η × η) * 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



##### 4.2-2 IsStemDomain
 ‣ IsStemDomain( G ) ( property )
 ‣ IsoclinicStemDomain( G ) ( attribute )
 ‣ AllStemGroupIds( n ) ( operation )
 ‣ AllStemGroupFamilies( n ) ( operation )

A group G is a stem group if Z(G) ≤ [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 ] ] ]



##### 4.2-3 IsoclinicRank
 ‣ IsoclinicRank( G ) ( attribute )
 ‣ IsoclinicMiddleLength( G ) ( attribute )

Let G be a finite p-group. Then log_p |[G,G] / (Z(G) ∩ [G,G])| is called the middle length of G. Also log_p |Z(G) ∩ [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 [JNO90].


gap> IsoclinicMiddleLength(G);
1
gap> IsoclinicRank(G);
4



#### 4.3 Isoclinism for crossed modules

##### 4.3-1 Isoclinism
 ‣ Isoclinism( X0, Y0 ) ( operation )
 ‣ AreIsoclinicDomains( X0, Y0 ) ( operation )

Let mathcalX,mathcalY be crossed modules with central quotients Q(mathcalX) and Q(mathcalY), and derived subcrossed modules [mathcalX,mathcalX] and [mathcalY,mathcalY] respectively. Let c_mathcalX : Q(mathcalX) × Q(mathcalX) -> [mathcalX,mathcalX] and c_mathcalY : Q(mathcalY) × Q(mathcalY) -> [mathcalY,mathcalY] be the two commutator maps. An isoclinism mathcalX ∼ mathcalY is a pair of bijective morphisms (η,ξ) where η : Q(mathcalX) -> Q(mathcalY) and ξ : [mathcalX,mathcalX] -> [mathcalY,mathcalY] such that c_mathcalX * ξ = (η × η) * c_mathcalY. Isoclinism is an equivalence relation, and all abelian crossed modules are isoclinic to the trivial crossed module.


gap> C8 := Cat1(16,8,1);;
gap> X8 := XMod(C8);  IdGroup( X8 );
[Group( [ f1*f2*f3, f3, f4 ] )->Group( [ f2, f2 ] )]
[ [ 8, 1 ], [ 2, 1 ] ]
gap> C9 := Cat1(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 ] )->Group( [ f2 ] )] => [Group( [ f1 ] )->Group( [ f2 ] )]],
[[Group( [ f3 ] )->Group( <identity> of ... )] => [Group(
[ f3 ] )->Group( <identity> of ... )]] ]



##### 4.3-2 IsStemDomain
 ‣ IsStemDomain( X0 ) ( property )
 ‣ IsoclinicStemDomain( X0 ) ( property )

A crossed module mathcalX is a stem crossed module if Z(mathcalX) ≤ [mathcalX,mathcalX]. 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



##### 4.3-3 IsoclinicRank
 ‣ IsoclinicRank( X0 ) ( attribute )
 ‣ IsoclinicMiddleLength( X0 ) ( attribute )

The formulae in subsection 4.2-3 are applied to the crossed module.


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


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