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Creation of a Cohomology Module

In order to compute the cohomology of a group with respect to a G-module M, it is first necessary to construct a data structure known as a cohomology module.

CohomologyModule(G, M) : GrpPerm, ModGrp -> ModCoho
CohomologyModule(G, M) : GrpPC, ModGrp -> ModCoho
CohomologyModule(G, M) : GrpMat, ModGrp -> ModCoho
CohomologyModule(G, M) : GrpFP, ModGrp -> ModCoho
Given a group G and a G-module M with acting group G this function returns a cohomology module for the action of G. The group G may be a finite permutation group, a finite matrix group, a PC-group, or any finitely presented group. For the PC-group case, however, the PC-presentation of G must be conditioned. This can be achieved by first executing the statement G := ConditionedGroup(G);
CohomologyModule(G, Q, T) : GrpPerm, SeqEnum, SeqEnum -> ModCoho
CohomologyModule(G, Q, T) : GrpPC, SeqEnum, SeqEnum -> ModCoho
CohomologyModule(G, Q, T) : GrpMat, SeqEnum, SeqEnum -> ModCoho
CohomologyModule(G, Q, T) : GrpFP, SeqEnum, SeqEnum -> ModCoho
Let G be a group which acts on a finitely-generated abelian group with invariants given by the sequence Q, and action described by T. The action T is given in the form of a sequence of d x d matrices over the integers, where d is the length of T, and T[i] defines the action of the i-th generator of G on the abelian group. The function returns a cohomology module for the action of G. The group G may be a finite permutation group, a finite matrix group, a PC-group or any finitely presented group. For the PC-group case, however, the PC-presentation of G must be conditioned. This can be achieved by first executing the statement G := ConditionedGroup(G);

Example GrpCoh_coho-module1 (H68E1)

We construct the cohomology module for PSL(3, 2) acting on a module of dimension 3 over GF(2). We first need to find a module of dimension 3.

> G := PSL(3, 2);
> Irrs := AbsolutelyIrreducibleModules(G, GF(2));
> Irrs;
[
    GModule of dimension 1 over GF(2),
    GModule of dimension 3 over GF(2),
    GModule of dimension 3 over GF(2),
    GModule of dimension 8 over GF(2)
]
> M := Irrs[2];
> CM := CohomologyModule(G, M);
> CM;
Cohomology Module

CohomologyModule(G, A, M) : GrpPerm, GrpAb, Any -> ModCoho
For a permutation group G acting on some abelian group A through M, compute the cohomology module. M has to be either a map from G into the endomorphisms of A, or a sequence of endomorphisms of A, one for each of the generators of G.
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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013