2 2d-objects

Here are the standard constructions which these implement:

• A conjugation crossed module is an 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 ¶: S ® R with central kernel, where r Î R acts on S by conjugation with ¶^-1r.
• 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 Î S to the inner automorphism of S by s.
• A trivial action crossed module ¶: S ® R has s^r = s for all s Î S,   r Î R, the source is abelian and the image lies in the centre of the range.
• An R-Module crossed module has an R-module as source and the zero map as boundary.
• The direct product X₁ ×X₂ of two crossed modules has source S₁ ×S₂, range R₁ ×R₂ and boundary ¶₁ ׶₂, with R₁, R₂ acting trivially on S₂, S₁ respectively.

In this implementation the attributes used in the construction of a crossed module X0 are:

• Source(X) and Range(X), the source S and range R of ¶ = Boundary(X);
• XModAction(X), a homomorphism from R to AutoGroup(X), a group of automorphisms of S:

More familiar attributes are Size and Name, formed by concatenating the names of the source and range. An Enumerator function has not yet been implemented.

Here is a simple example of an automorphism crossed module, the holomorph of the cyclic group of size five.

gap> c5 := Group( (1,2,3,4,5) );;
gap> SetName( c5, "c5" );
gap> X1 := XModByAutomorphismGroup( c5 );
[c5 -> PermAut(c5)] 
gap> Display( X1 );
Crossed module [c5 -> PermAut(c5)] :-
: Source group c5 has generators:
  [ (1,2,3,4,5) ]
: Range group PermAut(c5) has generators:
  [ (1,2,4,3) ]
: Boundary homomorphism maps source generators to:
  [ () ]
: Action homomorphism maps range generators to automorphisms:
  (1,2,4,3) --> { source gens --> [ (1,3,5,2,4) ] }
  This automorphism generates the group of automorphisms.
gap> Size( X1 );
[ 5, 4 ]

With the standard crossed module constructors listed above as building blocks, sub-crossed modules, quotients of normal sub-crossed modules, and also direct products may be constructed. A sub-crossed module S = (d: N ® M) is normal in X = (¶: S ® R) if

• N,M are normal subgroups of S,R respectively,
• d is the restriction of ¶,
• n^r Î N for all n Î N, r Î R,
• s^-1 s^m Î N for all m Î M, s Î S.

These conditions ensure that M \ltimes N is normal in the semidirect product R \ltimes S.

2.3 Pre-crossed modules

When axiom XMod 2 is not satisfied, the corresponding structure is known as a pre-crossed module. In this case the Peiffer subgroup of P of S is the subgroup of ker(¶) generated by Peiffer commutators

ás1,s2 ñ    =     (s1-1)¶s2 s2-1 s1 s2 ·

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

When both source and range groups are of the same type, corresponding properties are assigned to the crossed module.

In the following example the Peiffer subgroup ios cyclic of size 4.

gap> d16 := Group( (1,2,3,4,5,6,7,8), (2,8)(3,7)(4,6) );;
gap> SetName( d16, "d16" );
gap> gend16 := GeneratorsOfGroup( d16 );;
gap> sk4 := Subgroup( d16, [ (1,5)(3,7)(2,6)(4,8), (2,8)(3,7)(4,6) ] );;
gap> gensk4 := GeneratorsOfGroup( sk4 );;
gap> SetName( sk4, "sk4" );
gap> f16 := GroupHomomorphismByImages( d16, sk4, gend16, gensk4 );;
gap> P16 := PreXModByCentralExtension( f16 );;
gap> Display(P16);
Pre-crossed module [d16 -> sk4] :-
: Source group d16 has generators:
  [ (1,2,3,4,5,6,7,8), (2,8)(3,7)(4,6) ]
: Range group has generators:
  [ (1,5)(2,6)(3,7)(4,8), (2,8)(3,7)(4,6) ]
: Boundary homomorphism maps source generators to:
  [ (1,5)(2,6)(3,7)(4,8), (2,8)(3,7)(4,6) ]
: Action homomorphism maps range generators to automorphisms:
  (1,5)(2,6)(3,7)(4,8) --> { source gens -->
[ (1,2,3,4,5,6,7,8), (2,8)(3,7)(4,6) ] }
  (2,8)(3,7)(4,6) --> { source gens --> [ (1,8,7,6,5,4,3,2), (2,8)(3,7)(4,6) ] }
  These 2 automorphisms generate the group of automorphisms.

gap> P := PeifferSubgroup( P16 );
Group([ (1,7,5,3)(2,8,6,4) ])  
gap> X16 := XModByPeifferQuotient( P16 );;
gap> Display( X16 );
Crossed module [? -> sk4] :-
: Source group has generators:
  [ f1, f2 ]
: Range group has generators:
  [ (1,5)(2,6)(3,7)(4,8), (2,8)(3,7)(4,6) ]
: Boundary homomorphism maps source generators to:
  [ (2,8)(3,7)(4,6), (1,5)(2,6)(3,7)(4,8) ]
  The automorphism group is trivial
gap> iso16 := IsomorphismPermGroup( Source( X16 ) );;
gap> S16 := Image( iso16 );
Group([ (1,3)(2,4), (1,2)(3,4) ])   

2.4 Cat1-groups and pre-cat1-groups

In this implementation a cat1-group C has attributes:

In L1, Loday reformulated the notion of a crossed module as a cat1-group, namely a group G with a pair of homomorphisms t,h : G ® G having a common image R and satisfying certain axioms. We find it convenient to define a cat1-group C = (e;t,h : G ® R ) as having source group G, range group R, and three homomorphisms: two surjections t,h : G ® R and an embedding e : R ® G satisfying:

Cat 1   :   te  =  he  =  idR ,        Cat 2   :   [kert, kerh]  =  { 1G }  .

It follows that teh = h, het = t, tet = t, heh = h.

The maps t,h are often referred to as the source and target, but we choose to call them the tail and head of 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.

Here are some constructors for pre-cat1-groups and cat1-groups:

The following listing shows an example of a cat1-group of pc-groups:

gap> s3 := SymmetricGroup(IsPcGroup,3);;   SetName(s3,"s3");
gap> gens3 := GeneratorsOfGroup(s3);
[ f1, f2 ]
gap> c4 := CyclicGroup(4);;  SetName( c4, "c4" );
gap> s3c4 := DirectProduct( s3, c4 );;  SetName( s3c4, "s3c4" );  
gap> gens3c4 := GeneratorsOfGroup( s3c4 );
[ f1, f2, f3, f4 ]
gap> a := gens3[1];;  b := gens3[2];;  one := One(s3);;
gap> t2 := GroupHomomorphismByImages( s3c4, s3, gens3c4, [a,b,one,one] );
[ f1, f2, f3, f4 ] -> [ f1, f2, <identity> of ..., <identity> of ... ]
gap> e2 := Embedding( s3c4, 1 );
Pcgs([ f1, f2 ]) -> [ f1, f2 ]
gap> C2 := Cat1( t2, t2, e2 );
[s3c4=>s3]
gap> Display( C2 );
Cat1-group [s3c4=>s3] :-
: source group has generators:
  [ f1, f2, f3, f4 ]
:  range group has generators:
  [ f1, f2 ]
: tail homomorphism maps source generators to:
  [ f1, f2, <identity> of ..., <identity> of ... ]
: head homomorphism maps source generators to:
  [ f1, f2, <identity> of ..., <identity> of ... ]
: range embedding maps range generators to:
  [ f1, f2 ]
: kernel has generators:
  [ f3, f4 ]
: boundary homomorphism maps generators of kernel to:
  [ <identity> of ..., <identity> of ... ]
: kernel embedding maps generators of kernel to:
  [ f3, f4 ]
gap> IsPcCat1( C2 );
true
gap> Size( C2 );
[ 24, 6 ]

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 X = (¶: S ® R) the group G is defined as the semidirect product G = R \ltimes S using the action from X, with multiplication rule

(r1,s1) (r2,s2)  =  (r1 r2, s1r2 s2)  .

The structural morphisms are given by

t(r,s) = r,    h(r,s) = r (¶s),    er = (r,1)  .

On the other hand, starting with a cat1-group C = (e;t,h : G ® R), we define S = kert, the range R remains unchanged, and ¶ = h | _S . The action of R on S is conjugation in S via the embedding of R in G.

gap> X2 := XModOfCat1( C2 );
[Group( [ f3, f4 ] )->s3]
gap> Display( X2 );
Crossed module [..->s3] :-
: Source group has generators:
  [ f3, f4 ]
: Range group s3 has generators:
  [ f1, f2 ]
: Boundary homomorphism maps source generators to:
  [ <identity> of ..., <identity> of ... ]
  The automorphism group is trivial

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