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A short exact sequence of ZG-modules A >--> B -->> C induces a long exact sequence of cohomology groups --> Hn(G,A) --> Hn(G,B) --> Hn(G,C) --> Hn+1(G,A) --> . The implementation of this sequence is joint work with Daher Al-Baydli.
Consider the symmetric group G=S4 and the sequence Z/4Z >-----> Z/8Z ---> Z/2Z of trivial ZG-modules. We can represent a ZG-module as a GOuterGroup. The following commands use this representation to compute the induced cohomology homomorphism f: H3(S4,Z/4Z) ----> H3(S4,Z/8Z) and determine that the image of this induced homomorphism has order 8 and that its kernel has order 2.
gap> G:=SymmetricGroup(4);; gap> x:=(1,2,3,4,5,6,7,8);; gap> a:=Group(x^2);; gap> b:=Group(x);; gap> ahomb:=GroupHomomorphismByFunction(a,b,y->y);; gap> A:=TrivialGModuleAsGOuterGroup(G,a);; gap> B:=TrivialGModuleAsGOuterGroup(G,b);; gap> phi:=GOuterGroupHomomorphism();; gap> phi!.Source:=A;; gap> phi!.Target:=B;; gap> phi!.Mapping:=ahomb;; gap> Hphi:=CohomologyHomomorphism(phi,3);; gap> Size(ImageOfGOuterGroupHomomorphism(Hphi)); 8 gap> Size(KernelOfGOuterGroupHomomorphism(Hphi)); 2
The following commands then compute the homomorphism H3(S4,Z/8Z) ----> H3(S4,Z/2Z) induced by Z/4Z >-----> Z/8Z ---->> Z/2Z . and determine that the kernel of this homomorphsim has order 8.
gap> bhomc:=NaturalHomomorphismByNormalSubgroup(b,a); gap> B:=TrivialGModuleAsGOuterGroup(G,b); gap> C:=TrivialGModuleAsGOuterGroup(G,Image(bhomc)); gap> psi:=GOuterGroupHomomorphism(); gap> psi!.Source:=B; gap> psi!.Target:=C; gap> psi!.Mapping:=bhomc; gap> Hpsi:=CohomologyHomomorphism(psi,3); gap> Size(KernelOfGOuterGroupHomomorphism(Hpsi)); 8
The following commands then compute the connecting homomorphism H2(S4,Z/2Z) ----> H3(S4,Z/4Z) and determine that the image of this homomorphism has order 2.
gap> delta:=ConnectingCohomologyHomomorphism(psi,2);; gap> Size(ImageOfGOuterGroupHomomorphism(delta)); 2
Note that the various orders are consistent with exactness of the sequence H2(S4,Z/2Z) ----> H3(S4,Z/4Z) ----> H3(S4,Z/8Z) ----> H3(S4,Z/2Z)
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