2d-objects
Constructions for crossed modules
crossed module
2d-object
A crossed module \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}.

The kernel of \partial is abelian.

There are a variety of constructors for crossed modules:

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 \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 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 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.
In this implementation the attributes used in the construction of a crossed module X0 are as follows. Source(X0) and Range(X0) are the source S and range R of \partial, the boundary Boundary(X0); AutoGroup(X0) is a group of automorphisms of S; XModAction(X0) is a homomorphism from R to AutoGroup(X0).
More familiar attributes are Size and Name, the latter formed by concatenating the names of the source and range (if these exist). An Enumerator function has not been implemented. display a 2d-object The Display function is used to print details of 2d-objects.

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

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: [ (1,2,4,3) ] : Boundary homomorphism maps source generators to: [ () ] : Action homomorphism maps range generators to automorphisms: (1,2,4,3) --> { source gens --> [ (5,7,9,6,8) ] } This automorphism generates the group of automorphisms. gap> Size( X1 ); [ 5, 4 ] gap> Print( RepresentationsOfObject(X1), "\n" ); [ "IsComponentObjectRep", "IsAttributeStoringRep", "IsPreXModObj" ] gap> Print( KnownPropertiesOfObject(X1), "\n" ); [ "Is2dObject", "IsPerm2dObject", "IsPreXMod", "IsXMod", "IsTrivialAction2dObject", "IsAutomorphismGroup2dObject" ] gap> Print( KnownAttributesOfObject(X1), "\n" ); [ "Name", "Size", "Range", "Source", "Boundary", "AutoGroup", "XModAction" ]
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}\,s^m \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.
Pre-crossed modules
pre-crossed module
When axiom {\bf XMod\ 2} is not satisfied, the corresponding structure is known as a pre-crossed module.
gap> c := (11,12,13,14,15,16,17,18);; d := (12,18)(13,17)(14,16);; gap> d16 := Group( c, d );; gap> sk4 := Subgroup( d16, [ c^4, d ] );; gap> SetName( d16, "d16" ); SetName( sk4, "sk4" ); gap> bdy16 := GroupHomomorphismByImages( d16, sk4, [c,d], [c^4,d] );; gap> h1 := GroupHomomorphismByImages( d16, d16, [c,d], [c^5,d] );; gap> h2 := GroupHomomorphismByImages( d16, d16, [c,d], [c,c^4*d] );; gap> aut16 := Group( [ h1, h2 ] );; gap> act16 := GroupHomomorphismByImages( sk4, aut16, [c^4,d], [h1,h2] );; gap> P16 := PreXModByBoundaryAndAction( bdy16 ); [d16->sk4]
Peiffer subgroup
The Peiffer subgroup of a pre-crossed module P of S is the subgroup of {\rm ker}(\partial) generated by Peiffer commutators \langle s_1,s_2 \rangle \quad=\quad % (s_1^{-1})^{\partial s_2}\,s_2^{-1}\,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 ); [d16/P->sk4] gap> Display( X16 ); Crossed module [d16/P->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([ (1,3)(2,4), (1,2)(3,4) ])
When both source and range groups are of the same type, corresponding properties are assigned to the crossed module.
Cat1-groups and pre-cat1-groups
cat1-group
These are the attributes of a cat1-group \mathcal{C} in this implementation.

In , Loday reformulated the notion of a crossed module as a cat1-group, namely a group G with a pair of homomorphisms t,h : G \to G having a common image R and satisfying certain axioms. We find it 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} ~:~ te = he = {\rm id}_R, \qquad {\bf Cat\ 2} ~:~ [\ker t, \ker h] = \{ 1_G \}. 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 \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.
These are some of the 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);; gap> gens3 := GeneratorsOfGroup(s3); [ f1, f2 ] gap> pc4 := CyclicGroup(4);; gap> SetName(s3,"s3"); SetName( pc4, "pc4" ); gap> s3c4 := DirectProduct( s3, pc4 );; gap> 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, &lt;identity&gt; of ..., &lt;identity&gt; of ... ] gap> e2 := Embedding( s3c4, 1 ); [ 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, &lt;identity&gt; of ..., &lt;identity&gt; of ... ] : head homomorphism maps source generators to: [ f1, f2, &lt;identity&gt; of ..., &lt;identity&gt; of ... ] : range embedding maps range generators to: [ f1, f2 ] : kernel has generators: [ f3, f4 ] : boundary homomorphism maps generators of kernel to: [ &lt;identity&gt; of ..., &lt;identity&gt; 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 \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 remains unchanged, and \partial = h\!\mid_S . The action of R on S is conjugation in G via the embedding of R in G.
gap> SetName( Kernel(t2), "ker(t2)" );; gap> X2 := XModOfCat1( C2 ); [Group( [ f3, f4 ] )->s3] gap> Display( X2 ); Crossed module [ker(t2)->s3] :- : Source group has generators: [ f3, f4 ] : Range group s3 has generators: [ f1, f2 ] : Boundary homomorphism maps source generators to: [ &lt;identity&gt; of ..., &lt;identity&gt; of ... ] The automorphism group is trivial : associated cat1-group is [s3c4=>s3]
Selection of a small cat1-group
selection of a small cat1-group

The Cat1 function may also be used to select a cat1-group from a data file. All cat1-structures on groups of size up to 47 are stored in a list in file cat1data.g. Global variables CAT1&uscore;LIST&uscore;MAX&uscore;SIZE := 47 and CAT1&uscore;LIST&uscore;CLASS&uscore;SIZES are also stored. The &XMod;~2 version of the database orders the groups of size up to 47 according to the &GAP;~4 numbering of small groups. The data is read into the list CAT1&uscore;LIST only when this function is called.

This function may be used in three ways. Cat1Select( size ) returns the names of the groups with this size. Cat1Select( size, gpnum ) prints a list of cat1-structures for this chosen group. Cat1Select( size, gpnum, num ) (or just Cat1( 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> L18 := Cat1Select( 18 ); #I Loading cat1-group data into CAT1_LIST Usage: Cat1( size, gpnum, num ) [ "d18", "c18", "s3c3", "c3^2|Xc2", "c6c3" ] gap> Cat1Select( 18, 4 ); There are 4 cat1-structures for the group c3^2|Xc2. [ [range gens], source & range names, [tail genimages], [head genimages] ] :- [ [ (1,2,3), (4,5,6), (2,3)(5,6) ], tail = head = identity mapping ] [ [ (2,3)(5,6) ], "c3^2", "c2", [ (), (), (2,3)(5,6) ], [ (), (), (2,3)(5,6) ] ] [ [ (4,5,6), (2,3)(5,6) ], "c3", "s3", [ (), (4,5,6), (2,3)(5,6) ], [ (), (4,5,6), (2,3)(5,6) ] ] [ [ (4,5,6), (2,3)(5,6) ], "c3", "s3", [ (4,5,6), (4,5,6), (2,3)(5,6) ], [ (), (4,5,6), (2,3)(5,6) ] ] Usage: Cat1( size, gpnum, num ); Group has generators [ (1,2,3), (4,5,6), (2,3)(5,6) ] 4 gap> C4 := Cat1( 18, 4, 4 ); [c3^2|Xc2=>s3] gap> Display( C4 ); Cat1-group [c3^2|Xc2=>s3] :- : source group has generators: [ (1,2,3), (4,5,6), (2,3)(5,6) ] : range group has generators: [ (4,5,6), (4,5,6), (2,3)(5,6) ] : tail homomorphism maps source generators to: [ (4,5,6), (4,5,6), (2,3)(5,6) ] : head homomorphism maps source generators to: [ (), (4,5,6), (2,3)(5,6) ] : range embedding maps range generators to: [ (4,5,6), (4,5,6), (2,3)(5,6) ] : kernel has generators: [ ( 1, 2, 3)( 4, 6, 5) ] : boundary homomorphism maps generators of kernel to: [ (4,6,5) ] : kernel embedding maps generators of kernel to: [ (1,2,3)(4,6,5) ] gap> XC4 := XModOfCat1( C4 ); [Group( [ ( 1, 2, 3)( 4, 6, 5) ] )->s3]