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

Given a finite 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, matrix group or PC-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, invar, mats) : GrpPerm, SeqEnum, SeqEnum -> ModCoho

CohomologyModule(G, invar, mats) : GrpPC, SeqEnum, SeqEnum -> ModCoho

Let G be a finite group which acts on a finitely-generated abelian group with invariants given by the sequence invar, and action described by mats. The action mats is given in the form of a sequence of d x d matrices over the integers, where d is the length of invar, and mats[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, matrix group or PC-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 (H23E1)

> 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