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2 2d-groups : crossed modules and cat1-groups

The term 2d-group refers to a set of equivalent categories of which the most common are the categories of crossed modules; cat1-groups; and group-groupoids, all of which involve a pair of groups.

2.1 Constructions for crossed modules

A crossed module (of groups) \(\mathcal{X} = (\partial : S \to R )\) consists of a group homomorphism \(\partial \), called the boundary of \(\mathcal{X}\), with source \(S\) and range \(R\). The group \(R\) acts on itself by conjugation, and on \(S\) by an action \(\alpha : R \to {\rm Aut}(S)\) such that, for all \(s,s_1,s_2 \in S\) and \(r \in R\),

\[ {\bf XMod\ 1} : \partial(s^r) = r^{-1} (\partial s) r = (\partial s)^r, \qquad {\bf XMod\ 2} : s_1^{\partial s_2} = s_2^{-1}s_1 s_2 = {s_1}^{s_2}. \]

When only the first of these axioms is satisfied, the resulting structure is a pre-crossed module (see section 2.3). (Much of the literature on crossed modules uses left actions, but we have chosen to use right actions in this package since that is the standard choice for group actions in GAP.)

The kernel of \(\partial\) is abelian.

There are a variety of constructors for crossed modules:

2.1-1 XMod

The global function XMod implements one of the following standard constructions:

• A trivial action crossed module \((\partial : S \to R)\) has \(s^r = s\) for all \(s \in S, \; r \in R\), the source is abelian and the image lies in the centre of the range.

• A conjugation crossed module is the inclusion of a normal subgroup \(S \unlhd R\), where \(R\) acts on \(S\) by conjugation.

• A central extension crossed module has as boundary a surjection \(\partial : S \to R\), with central kernel, where \(r \in R\) acts on \(S\) by conjugation with \(\partial^{-1}r\).

• An automorphism crossed module has as range a subgroup \(R\) of the automorphism group Aut\((S)\) of \(S\) which contains the inner automorphism group of \(S\). The boundary maps \(s \in S\) to the inner automorphism of \(S\) by \(s\).

• A crossed abelian module has an abelian module as source and the zero map as boundary.

• The direct product \(\mathcal{X}_{1} \times \mathcal{X}_{2}\) of two crossed modules has source \(S_1 \times S_2\), range \(R_1 \times R_2\) and boundary \(\partial_1 \times \partial_2\), with \(R_1,\ R_2\) acting trivially on \(S_2,\ S_1\) respectively.

  Since DirectProduct is a global function which only accepts groups, it is necessary to provide an "other method" for operation DirectProductOp which, as usual, takes as parameters a list of crossed modules, followed by the first of these: DirectProductOp([X1,X2],X1);

2.1-2 Source

The following attributes are used in the construction of a crossed module X0.

• Source(X0) and Range(X0) are the source \(S\) and range \(R\) of \(\partial\), the boundary Boundary(X0);

• XModAction(X0) is a homomorphism from \(R\) to a group of automorphisms of X0.

(Up until version 2.63 there was an additional attribute AutoGroup, the range of XModAction(X0).)

2.1-3 ImageElmXModAction

This function returns the element \(s^r\) given by XModAction(X0).

2.1-4 Size

More familiar attributes are Name, Size and IdGroup. The name is formed by concatenating the names of the source and range (if these exist). Size and IdGroup return two-element lists.

The ExternalSetXMod for a crossed module is the source group considered as a G-set of the range group using the crossed module action.

The Display function is used to print details of 2d-groups.

In the simple example below, X1 is an automorphism crossed module, using a cyclic group of size five. The Print statements at the end list the GAP representations, properties and attributes of X1.

gap> c5 := Group( (5,6,7,8,9) );;
gap> SetName( c5, "c5" );
gap> X1 := XModByAutomorphismGroup( c5 );
[c5 -> PAut(c5)] 
gap> Display( X1 );
Crossed module [c5 -> PAut(c5)] :-
: Source group c5 has generators:
  [ (5,6,7,8,9) ]
: Range group PAut(c5) has generators:
  [ (1,2,3,4) ]
: Boundary homomorphism maps source generators to:
  [ () ]
: Action homomorphism maps range generators to automorphisms:
  (1,2,3,4) --> { source gens --> [ (5,7,9,6,8) ] }
  This automorphism generates the group of automorphisms.
gap> Size( X1 );  IdGroup( X1 ); 
[ 5, 4 ]
[ [ 5, 1 ], [ 4, 1 ] ]
gap> ext := ExternalSetXMod( X1 ); 
<xset:[ (), (5,6,7,8,9), (5,7,9,6,8), (5,8,6,9,7), (5,9,8,7,6) ]>
gap> Orbits( ext );
[ [ () ], [ (5,6,7,8,9), (5,7,9,6,8), (5,9,8,7,6), (5,8,6,9,7) ] ]
gap> ImageElmXModAction( X1, (5,6,7,8,9), (1,2,3,4) );
(5,7,9,6,8)
gap> RepresentationsOfObject( X1 );
[ "IsComponentObjectRep", "IsAttributeStoringRep", "IsPreXModObj" ]
gap> KnownAttributesOfObject( X1);
[ "Name", "Size", "Range", "Source", "IdGroup", "Boundary", "XModAction", 
  "ExternalSetXMod" ]

2.2 Properties of crossed modules

The underlying category structures for the objects constructed in this chapter follow the sequence Is2DimensionalDomain; Is2DimensionalMagma; Is2DimensionalMagmaWithOne; Is2DimensionalMagmaWithInverses, mirroring the situation for (one-dimensional) groups. From these we construct Is2DimensionalSemigroup, Is2DimensionalMonoid and Is2DimensionalGroup.

There are then a variety of properties associated with crossed modules, starting with IsPreXMod and IsXMod.

2.2-1 IsXMod

A structure which has IsPerm2DimensionalGroup is a precrossed module or a pre-cat1-group (see section 2.4) whose source and range are both permutation groups. The properties IsPc2DimensionalGroup, IsFp2DimensionalGroup are defined similarly. In the example below we see that X1 has IsPreXMod, IsXMod and IsPerm2DimensionalGroup. There are also properties corresponding to the various construction methods listed in section 2.1: IsTrivialAction2DimensionalGroup; IsNormalSubgroup2DimensionalGroup; IsCentralExtension2DimensionalGroup; IsAutomorphismGroup2DimensionalGroup; IsAbelianModule2DimensionalGroup.

gap> KnownPropertiesOfObject( X1 );
[ "IsEmpty", "IsTrivial", "IsNonTrivial", "IsFinite", 
  "CanEasilyCompareElements", "CanEasilySortElements", "IsDuplicateFree", 
  "IsGeneratorsOfSemigroup", "IsPreXModDomain", "IsPerm2DimensionalGroup", 
  "IsPreXMod", "IsXMod", "IsAutomorphismGroup2DimensionalGroup" ]

2.2-2 SubXMod

With the standard crossed module constructors listed above as building blocks, sub-crossed modules, normal sub-crossed modules \(\mathcal{N} \lhd \mathcal{X}\), and also quotients \(\mathcal{X}/\mathcal{N}\) may be constructed. A sub-crossed module \(\mathcal{S} = (\delta : N \to M)\) is normal in \(\mathcal{X} = (\partial : S \to R)\) if

• \(N,M\) are normal subgroups of \(S,R\) respectively,

• \(\delta\) is the restriction of \(\partial\),

• \(n^r \in N\) for all \(n \in N,~r \in R\),

• \((s^{-1})^ms \in N\) for all \(m \in M,~s \in S\).

These conditions ensure that \(M \ltimes N\) is normal in the semidirect product \(R \ltimes S\). (Note that \(\langle s,m \rangle = (s^{-1})^ms\) is a displacement: see Displacement (4.1-3).)

A method for IsNormal for precrossed modules is provided. See section 4.1 for factor crossed modules and their natural morphisms.

The five normal subcrossed modules of X4 found in the following example are [id,id], [k4,k4], [k4,a4], [a4,a4] and X4 itself.

gap> s4 := Group( (1,2), (2,3), (3,4) );; 
gap> a4 := Subgroup( s4, [ (1,2,3), (2,3,4) ] );; 
gap> k4 := Subgroup( a4, [ (1,2)(3,4), (1,3)(2,4) ] );; 
gap> SetName(s4,"s4");  SetName(a4,"a4");  SetName(k4,"k4"); 
gap> X4 := XModByNormalSubgroup( s4, a4 );
[a4->s4]
gap> Y4 := SubXMod( X4, k4, a4 ); 
[k4->a4]
gap> IsNormal(X4,Y4);
true
gap> NX4 := NormalSubXMods( X4 );;
gap> Length( NX4 ); 
5

2.3 Pre-crossed modules

2.3-1 PreXModByBoundaryAndAction

If axiom \({\bf XMod\ 2}\) is not satisfied, the corresponding structure is known as a pre-crossed module.

gap> b1 := (11,12,13,14,15,16,17,18);;  b2 := (12,18)(13,17)(14,16);;
gap> d16 := Group( b1, b2 );;
gap> sk4 := Subgroup( d16, [ b1^4, b2 ] );;
gap> SetName( d16, "d16" );  SetName( sk4, "sk4" );
gap> bdy16 := GroupHomomorphismByImages( d16, sk4, [b1,b2], [b1^4,b2] );;
gap> aut1 := GroupHomomorphismByImages( d16, d16, [b1,b2], [b1^5,b2] );;
gap> aut2 := GroupHomomorphismByImages( d16, d16, [b1,b2], [b1,b2^4*b2] );;
gap> aut16 := Group( [ aut1, aut2 ] );;
gap> act16 := GroupHomomorphismByImages( sk4, aut16, [b1^4,b2], [aut1,aut2] );;
gap> P16 := PreXModByBoundaryAndAction( bdy16, act16 );
[d16->sk4]
gap> IsXMod(P16);
false

2.3-2 PeifferSubgroup

The Peiffer subgroup \(P\) of a pre-crossed module \({\mathcal X}\) is the subgroup of \({\rm ker}(\partial)\) generated by Peiffer commutators

\[ \lfloor s_1,s_2 \rfloor ~=~ (s_1^{-1})^{\partial s_2}~s_2^{-1}~s_1~s_2 ~=~ \langle \partial s_2, s_1 \rangle\ [s_1,s_2]~. \]

Then \(\mathcal{P} = (0 : P \to \{1_R\})\) is a normal sub-pre-crossed module of \(\mathcal{X}\) and \(\mathcal{X}/\mathcal{P} = (\partial : S/P \to R)\) is a crossed module.

In the following example the Peiffer subgroup is cyclic of size \(4\).

gap> P := PeifferSubgroup( P16 );
Group( [ (11,15)(12,16)(13,17)(14,18), (11,17,15,13)(12,18,16,14) ] )
gap> X16 := XModByPeifferQuotient( P16 );
Peiffer([d16->sk4])
gap> Display( X16 );
Crossed module Peiffer([d16->sk4]) :-
: Source group has generators:
  [ f1, f2 ]
: Range group has generators:
  [ (11,15)(12,16)(13,17)(14,18), (12,18)(13,17)(14,16) ]
: Boundary homomorphism maps source generators to:
  [ (12,18)(13,17)(14,16), (11,15)(12,16)(13,17)(14,18) ]
  The automorphism group is trivial
gap> iso16 := IsomorphismPermGroup( Source( X16 ) );;
gap> S16 := Image( iso16 );
Group([ (1,2), (3,4) ])   

2.4 Cat1-groups and pre-cat1-groups

In [Lod82], Loday reformulated the notion of a crossed module as a cat1-group, namely a group \(G\) with a pair of endomorphisms \(t,h : G \to G\) having a common image \(R\) and satisfying certain axioms. We find it computationally convenient to define a cat1-group \(\mathcal{C} = (e;t,h : G \to R )\) as having source group \(G\), range group \(R\), and three homomorphisms: two surjections \(t,h : G \to R\) and an embedding \(e : R \to G\) satisfying:

\[ {\bf Cat\ 1} : ~t \circ e ~=~ h \circ e = {\rm id}_R, \qquad {\bf Cat\ 2} : ~[\ker t, \ker h] ~=~ \{ 1_G \}. \]

It follows that \(\;t \circ e \circ h = h,~ h \circ e \circ t = t,~ t \circ e \circ t = t~\) and \(~h \circ e \circ h = h\). (See section 2.5 for the case when \(t,h\) are endomorphisms.)

2.4-1 Cat1Group

The global functions Cat1Group and PreCat1Group can be called in various ways.

• as Cat1Group(t,h,e); when \(t,h,e\) are three homomorphisms, which is equivalent to PreCat1GroupByTailHeadEmbedding(t,h,e);

• as Cat1Group(t,h); when \(t,h\) are two endomorphisms, which is equivalent to PreCat1GroupByEndomorphisms(t,h);

• as Cat1Group(t); when \(t=h\) is an endomorphism, which is equivalent to PreCat1GroupByEndomorphisms(t,t);

• as Cat1Group(t,e); when \(t=h\) and \(e\) are homomorphisms, which is equivalent to PreCat1GroupByTailHeadEmbedding(t,t,e);

• as Cat1Group(i,j,k); when \(i,j,k\) are integers, which is equivalent to Cat1Select(i,j,k); as described in section 2.6.

gap> g18gens := [ (1,2,3), (4,5,6), (2,3)(5,6) ];;     
gap> s3agens := [ (7,8,9), (8,9) ];;                
gap> g18 := Group( g18gens );;  SetName( g18, "g18" ); 
gap> s3a := Group( s3agens );;  SetName( s3a, "s3a" );
gap> t1 := GroupHomomorphismByImages(g18,s3a,g18gens,[(7,8,9),(),(8,9)]);     
[ (1,2,3), (4,5,6), (2,3)(5,6) ] -> [ (7,8,9), (), (8,9) ]
gap> h1 := GroupHomomorphismByImages(g18,s3a,g18gens,[(7,8,9),(7,8,9),(8,9)]);
[ (1,2,3), (4,5,6), (2,3)(5,6) ] -> [ (7,8,9), (7,8,9), (8,9) ]
gap> e1 := GroupHomomorphismByImages(s3a,g18,s3agens,[(1,2,3),(2,3)(5,6)]);   
[ (7,8,9), (8,9) ] -> [ (1,2,3), (2,3)(5,6) ]
gap> C18 := Cat1Group( t1, h1, e1 );
[g18=>s3a]

2.4-2 Source

These are the attributes of a cat1-group \(\mathcal{C}\) in this implementation.

The maps \(t,h\) are often referred to as the source and target, but we choose to call them the tail and head of \(\mathcal{C}\), because source is the GAP term for the domain of a function. The RangeEmbedding is the embedding of R in G, the KernelEmbedding is the inclusion of the kernel of t in G, and the Boundary is the restriction of h to the kernel of t. It is frequently the case that \(t=h\), but not in the example C18 above.

gap> Source( C18 );
g18
gap> Range( C18 );
s3a
gap> TailMap( C18 );
[ (1,2,3), (4,5,6), (2,3)(5,6) ] -> [ (7,8,9), (), (8,9) ]
gap> HeadMap( C18 );
[ (1,2,3), (4,5,6), (2,3)(5,6) ] -> [ (7,8,9), (7,8,9), (8,9) ]
gap> RangeEmbedding( C18 );
[ (7,8,9), (8,9) ] -> [ (1,2,3), (2,3)(5,6) ]
gap> Kernel( C18 );
Group([ (4,5,6) ])
gap> KernelEmbedding( C18 );
[ (4,5,6) ] -> [ (4,5,6) ]
gap> Name( C18 );
"[g18=>s3a]"
gap> Size( C18 );
[ 18, 6 ]
gap> StructureDescription( C18 );
[ "(C3 x C3) : C2", "S3" ]

2.4-3 DiagonalCat1Group

These are some more constructors for cat1-groups. The following listing shows an example of a permutation cat1-group of size \([576,24]\) with source group \(S_4 \times S_4\), range group a third \(S_4\), and \(t \neq h\). A similar example may be reproduced using the command C := DiagonalCat1Group([(1,2,3,4),(3,4)]);.

gap> G4 := Group( (1,2,3,4), (3,4), (5,6,7,8), (7,8) );; 
gap> R4 := Group( (9,10,11,12), (11,12) );;
gap> SetName( G4, "s4s4" );  SetName( R4, "s4d" ); 
gap> G4gens := GeneratorsOfGroup( G4 );; 
gap> R4gens := GeneratorsOfGroup( R4 );; 
gap> t := GroupHomomorphismByImages( G4, R4, G4gens, 
>            Concatenation( R4gens, [ (), () ] ) );; 
gap> h := GroupHomomorphismByImages( G4, R4, G4gens,  
>            Concatenation( [ (), () ], R4gens ) );; 
gap> e := GroupHomomorphismByImages( R4, G4, R4gens, 
>            [ (1,2,3,4)(5,6,7,8), (3,4)(7,8) ] );; 
gap> C4 := PreCat1GroupByTailHeadEmbedding( t, h, e );; 
gap> Display(C4);
Cat1-group [s4s4=>s4d] :- 
: Source group s4s4 has generators:
  [ (1,2,3,4), (3,4), (5,6,7,8), (7,8) ]
: Range group s4d has generators:
  [ ( 9,10,11,12), (11,12) ]
: tail homomorphism maps source generators to:
  [ ( 9,10,11,12), (11,12), (), () ]
: head homomorphism maps source generators to:
  [ (), (), ( 9,10,11,12), (11,12) ]
: range embedding maps range generators to:
  [ (1,2,3,4)(5,6,7,8), (3,4)(7,8) ]
: kernel has generators:
  [ (5,6,7,8), (7,8) ]
: boundary homomorphism maps generators of kernel to:
  [ ( 9,10,11,12), (11,12) ]
: kernel embedding maps generators of kernel to:
  [ (5,6,7,8), (7,8) ]

2.5 Properties of cat1-groups and pre-cat1-groups

Many of the properties listed in section 2.2 apply to pre-cat1-groups and to cat1-groups since these are also 2d-groups. There are also more specific properties.

2.5-1 IsCat1Group

IsIdentityCat1Group(C0) is true when the head and tail maps of C0 are identity mappings. IsEndomorphismPreCat1Group(C0) is true when the range of C0 is a subgroup of the source. When this is not the case, replacing \(t,h,e\) by \(t*e,h*e\) and the inclusion mapping of the image of \(e\) gives an isomorphic cat1-group for which IsEndomorphismPreCat1Group is true.

gap> G2 := SmallGroup( 288, 956 );  SetName( G2, "G2" );
<pc group of size 288 with 7 generators>
gap> d12 := DihedralGroup( 12 );  SetName( d12, "d12" );
<pc group of size 12 with 3 generators>
gap> a1 := d12.1;;  a2 := d12.2;;  a3 := d12.3;;  a0 := One( d12 );;
gap> gensG2 := GeneratorsOfGroup( G2 );;
gap> t2 := GroupHomomorphismByImages( G2, d12, gensG2,
>           [ a0, a1*a3, a2*a3, a0, a0, a3, a0 ] );;
gap> h2 := GroupHomomorphismByImages( G2, d12, gensG2,
>           [ a1*a2*a3, a0, a0, a2*a3, a0, a0, a3^2 ] );;                   
gap> e2 := GroupHomomorphismByImages( d12, G2, [a1,a2,a3],
>        [ G2.1*G2.2*G2.4*G2.6^2, G2.3*G2.4*G2.6^2*G2.7, G2.6*G2.7^2 ] );
[ f1, f2, f3 ] -> [ f1*f2*f4*f6^2, f3*f4*f6^2*f7, f6*f7^2 ]
gap> C2 := PreCat1GroupByTailHeadEmbedding( t2, h2, e2 );
[G2=>d12]
gap> IsCat1Group( C2 );
true
gap> KnownPropertiesOfObject( C2 );
[ "CanEasilyCompareElements", "CanEasilySortElements", "IsDuplicateFree", 
  "IsGeneratorsOfSemigroup", "IsPreCat1Domain", "IsPerm2DimensionalGroup", 
  "IsPreCat1Group", "IsCat1Group", "IsEndomorphismPreCat1Group" ]
gap> IsEndomorphismPreCat1Group( C2 );
false
gap> EC4 := EndomorphismPreCat1Group( C4 );
[s4s4=>Group( [ (1,2,3,4)(5,6,7,8), (3,4)(7,8), (), () ] )]

2.5-2 Cat1GroupOfXMod

The category of crossed modules is equivalent to the category of cat1-groups, and the functors between these two categories may be described as follows. Starting with the crossed module \(\mathcal{X} = (\partial : S \to R)\) the group \(G\) is defined as the semidirect product \(G = R \ltimes S\) using the action from \(\mathcal{X}\), with multiplication rule

\[ (r_1,s_1)(r_2,s_2) ~=~ (r_1r_2,{s_1}^{r_2}s_2). \]

The structural morphisms are given by

\[ t(r,s) = r, \quad h(r,s) = r (\partial s), \quad er = (r,1). \]

On the other hand, starting with a cat1-group \( \mathcal{C} = (e;t,h : G \to R)\), we define \( S = \ker t\), the range \(R\) is unchanged, and \( \partial = h\!\mid_S \). The action of \(R\) on \(S\) is conjugation in \(G\) via the embedding of \(R\) in \(G\).

gap> X2 := XModOfCat1Group( C2 );;
gap> Display( X2 );

Crossed module X([G2=>d12]) :- 
: Source group has generators:
  [ f1, f4, f5, f7 ]
: Range group d12 has generators:
  [ f1, f2, f3 ]
: Boundary homomorphism maps source generators to:
  [ f1*f2*f3, f2*f3, <identity> of ..., f3^2 ]
: Action homomorphism maps range generators to automorphisms:
  f1 --> { source gens --> [ f1*f5, f4*f5, f5, f7^2 ] }
  f2 --> { source gens --> [ f1*f5*f7^2, f4, f5, f7 ] }
  f3 --> { source gens --> [ f1*f7, f4, f5, f7 ] }
  These 3 automorphisms generate the group of automorphisms.
: associated cat1-group is [G2=>d12]

gap> StructureDescription(X2);
[ "D24", "D12" ]

2.6 Selection of a small cat1-group

The Cat1Group function may also be used to select a cat1-group from a data file. All cat1-structures on groups of size up to \(70\) (ordered according to the GAP 4 numbering of small groups) are stored in a list in file cat1data.g. Global variables CAT1_LIST_MAX_SIZE := 70 and CAT1_LIST_CLASS_SIZES are also stored. The data is read into the list CAT1_LIST only when this function is called.

2.6-1 Cat1Select

The function Cat1Select may be used in three ways. Cat1Select( size ) returns the names of the groups with this size, while Cat1Select( size, gpnum ) prints a list of cat1-structures for this chosen group. Cat1Select( size, gpnum, num ) returns the chosen cat1-group.

The example below is the first case in which \(t \neq h\) and the associated conjugation crossed module is given by the normal subgroup c3 of s3.

gap> ## check the number of groups of size 18
gap> L18 := Cat1Select( 18 ); 
Usage:  Cat1Select( size, gpnum, num );
[ "D18", "C18", "C3 x S3", "(C3 x C3) : C2", "C6 x C3" ]
gap> ## check the number of cat1-structures on the fourth of these
gap> Cat1Select( 18, 4 );
Usage:  Cat1Select( size, gpnum, num );
There are 4 cat1-structures for the group (C3 x C3) : C2.
Using small generating set [ f1, f2, f2*f3 ] for source of homs.
[ [range gens], [tail genimages], [head genimages] ] :-
(1)  [ [ f1 ], [ f1, <identity> of ..., <identity> of ... ], 
  [ f1, <identity> of ..., <identity> of ... ] ]
(2)  [ [ f1, f3 ], [ f1, <identity> of ..., f3 ], 
  [ f1, <identity> of ..., f3 ] ]
(3)  [ [ f1, f3 ], [ f1, <identity> of ..., f3 ], 
  [ f1, f3^2, <identity> of ... ] ]
(4)  [ [ f1, f2, f2*f3 ],  tail = head = identity mapping ]
4
gap> ## select the third of these cat1-structures 
gap> C18 := Cat1Group( 18, 4, 3 );
[(C3 x C3) : C2=>Group( [ f1, <identity> of ..., f3 ] )]
gap> ## convert from a pc-cat1-group to a permutation cat1-group
gap> iso18 := IsomorphismPermObject( C18 );;
gap> PC18 := Image( iso18 );;
gap> Display( PC18 );
Cat1-group :- 
: Source group has generators:
  [ (2,3)(5,6), (4,5,6), (1,2,3) ]
: Range group has generators:
  [ (2,3), (), (1,2,3) ]
: tail homomorphism maps source generators to:
  [ (2,3), (), (1,2,3) ]
: head homomorphism maps source generators to:
  [ (2,3), (1,3,2), (1,2,3) ]
: range embedding maps range generators to:
  [ (2,3)(5,6), (), (1,2,3) ]
: kernel has generators:
  [ (4,5,6) ]
: boundary homomorphism maps generators of kernel to:
  [ (1,3,2) ]
: kernel embedding maps generators of kernel to:
  [ (4,5,6) ]
gap> convert the result to the associated permutation crossed module 
gap> X18 := XModOfCat1Group( PC18 );; 
gap> Display( X18 ); 
Crossed module:- 
: Source group has generators:
  [ (4,5,6) ]
: Range group has generators:
  [ (2,3), (), (1,2,3) ]
: Boundary homomorphism maps source generators to:
  [ (1,3,2) ]
: Action homomorphism maps range generators to automorphisms:
  (2,3) --> { source gens --> [ (4,6,5) ] }
  () --> { source gens --> [ (4,5,6) ] }
  (1,2,3) --> { source gens --> [ (4,5,6) ] }
  These 3 automorphisms generate the group of automorphisms.
: associated cat1-group is [..=>..]

2.6-2 AllCat1DataGroupsBasic

For a group \(G\) of size greater than \(70\) which is reasonably straightforward this function may be used to construct a list of all cat1-group structures on \(G\). The operation also attempts to write output to a file in the folder xmod/lib. (Other operations in the file cat1data.gi have been used to deal with the more complicated groups of size up to \(70\), but these are not described here.)

Van Luyen Le has a more efficient algorithm, extending the data up to groups of size 171, which is expected to appear in a future release of HAP.

gap> gp := SmallGroup( 102, 2 ); 
<pc group of size 102 with 3 generators>
gap> StructureDescription( gp ); 
"C3 x D34"
gap> all := AllCat1DataGroupsBasic( gp );
#I Edit last line of .../xmod/lib/nn.kk.out to end with ] ] ] ] ]
[ [Group( [ f1, f2, f3 ] )=>Group( [ f1, <identity> of ..., <identity> of ... 
     ] )], [Group( [ f1, f2, f3 ] )=>Group( [ f1, f2, <identity> of ... ] )], 
  [Group( [ f1, f2, f3 ] )=>Group( [ f1, <identity> of ..., f3 ] )], 
  [Group( [ f1, f2, f3 ] )=>Group( [ f1, f2, f3 ] )] ]

2.7 More functions for crossed modules and cat1-groups

Chapter 4 contains functions for quotient crossed modules; centre of a crossed module; commutator and derived subcrossed modules; etc.

Here we mention two functions for groups which have been extended to the two-dimensional case.

2.7-1 IdGroup

These functions return two-element lists formed by applying the function to the source and range of the 2d-group.

gap> IdGroup( X2 );
[ [ 24, 6 ], [ 12, 4 ] ]
gap> StructureDescription( C2 );
[ "(S3 x D24) : C2", "D12" ]

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