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### Some classical theory

The second cohomology H2(G,A) of G with coefficients in a G-module A, together with the corresponding 2-cocycles, have a useful group-theoretic interpretation.

Any group extension N >---> E --->> G gives rise to

1. an outer action alpha:G ---> Out(N) of G on N.
2. an action G ---> Aut(Z(N)) of G on the centre of N, uniquely induced by the outer action alpha and the canonical action of Out(N) on Z(N).
3. a 2-cocycle f:G×G ---> Z(N) with values in the G-module A=Z(N).

Any outer homomorphism alpha:G--->Out(N) gives rise to a cohomology class [c] in H3(G,Z(N)).

It was shown by Eilenberg and Mac Lane that the cohomology class [c] is trivial if and only if the outer action alpha arises from some group extension N >---> E --->> G.

If [c] is trivial then there is a bijection between the second cohomology group H2(G,Z(N)) and Yoneda equivalence classes of extensions of G by N that are compatible with alpha.

### An example

Consider the group H=SmallGroup(64,134). This is the Sylow 2-subgroup of the Mathieu group M12. Consider the normal subgroup N:=NormalSubgroups(G)[16] and quotient group G=H/N. We have N=C2×D4 , A=Z(N)=C2×C2 and G=C2×C2 .
Suppose that we wish to classify all extensions C2×D4 >---> E ---> C2×C2 which induce the given outer action of G on N. The following commands show that, up to Yoneda equivalence, there are two such extensions.

gap> H:=SmallGroup(64,134);; gap> N:=NormalSubgroups(H)[16];; gap> A:=Centre(GOuterGroup(H,N));; gap> G:=ActingGroup(A);; gap> R:=ResolutionFiniteGroup(G,3);; gap> C:=HomToGModule(R,A);; gap> Cohomology(C,2); [ 2 ]
The following additional commands return a standard 2-cocycle f:G×G-->A =Z2×Z2 corresponding to the non-trivial element in H2(G,A). The value f(g,h) of the 2-cocycle is calculated for all 16 pairs g,h in G.

gap> CH:=CohomologyModule(C,2);;
gap> Elts:=Elements(ActedGroup(CH));
[ <identity> of ..., f1 ]
gap> x:=Elts[2];;
gap> c:=CH!.representativeCocycle(x);
Standard 2-cocycle
gap> f:=Mapping(c);;
gap> for g in G do for h in G do
> Print(f(g,h),"\n");
> od;
> od;
<identity> of ...
<identity> of ...
<identity> of ...
<identity> of ...
<identity> of ...
f6
<identity> of ...
f6
<identity> of ...
<identity> of ...
<identity> of ...
<identity> of ...
<identity> of ...
f6
<identity> of ...
f6

The HAPcocyclic package developed by Robert Morse can be used to construct an extension of G by N for each cohomology class in H2(G,A). An initial developmental version of this package can be downloaded and untarred in directory ../gap/pkg . The following commands will then construct and identify all extensions of N by G corresponding to the given outer action of G on N.

gap> #The following commands only work if the HAPcocyclic package by Robert Morse is loaded.
gap> LoadPackage("HAPcocyclic");;
gap> H := SmallGroup(64,134);;
gap> N := NormalSubgroups(H)[16];;
gap> ON := GOuterGroup(H,N);;
gap> A := Centre(ON);;
gap> G:=ActingGroup(A);;
gap> R:=ResolutionFiniteGroup(G,3);;
gap> C:=HomToGModule(R,A);;
gap> CH:=CohomologyModule(C,2);;
gap> Elts:=Elements(ActedGroup(CH));;
gap> lst := List(Elts{[1..Length(Elts)]},x->CH!.representativeCocycle(x));;
gap> ccgrps := List(lst, x->CcGroup(ON, x));;
gap> #So ccgrps is a list of groups, each being an extension of G by N, corresponding
gap> #to the two elements in H^2(G,A).
gap> #The following command produces the GAP identification number for each group.
gap> L:=List(ccgrps,IdGroup);
[ [ 64, 134 ], [ 64, 135 ] ]

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