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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 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 ---> N with values in the group 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×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 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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