AboutHap

About HAP: Counting extensions of groups
(& computing extensions using HAPcocyclic)

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

an outer action alpha:G ---> Out(N) of G on N.
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).
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)[15];;

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×Z₂corresponding 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 following extended example illustrates how to construct a cohomology class C in H²(G, A) from a cocycle c:G x G --> A.

gap> #We'll construct G=SL(2,Z_4) as a permutation group.
gap> G:=SL(2,ZmodnZ(4));;
gap> G:=Image(IsomorphismPermGroup(G));;

gap> #We'll construct Z_8=Z/8Z as a G-outer group
gap> z_8:=Group((1,2,3,4,5,6,7,8));;
gap> Z_8:=TrivialGModuleAsGOuterGroup(G,z_8);;

gap> #We'll compute the group h=H^2(G,Z_8)
gap> R:=ResolutionFiniteGroup(G,3);; #R is a free resolution
gap> C:=HomToGModule(R,Z_8);; # C is a chain complex
gap> H:=CohomologyModule(C,2);; #H is the second cohomology H^2(G,Z_8)
gap> h:=ActedGroup(H);; #h is the underlying group of H

gap> #We'll compute cocycles c2, c5 for the second and fifth cohomology classs
gap> c2:=H!.representativeCocycle(Elements(h)[2]);
Standard 2-cocycle

gap> c5:=H!.representativeCocycle(Elements(h)[5]);
Standard 2-cocycle

gap> #Now we'll construct the cohomology classes C2, C5 in the group h corresponding to the cocycles c2, c5.
gap> C2:=CohomologyClass(H,c2);;
gap> C5:=CohomologyClass(H,c5);;

gap> #Finally, we'll show that C2, C5 are distinct cohomology classes, both of order 4.
gap> C2=C5;
false
gap> Order(C2);
4
gap> Order(C5);
4

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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