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Some classical theoryThe 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
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. |
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An exampleConsider 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. |
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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 ] |
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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 |
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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. |
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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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