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);

> 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

> G := ASL(3,5);              
> ChiefFactors(G);
    G
    |  A(2, 5)                    = L(3, 5)
    *
    |  Cyclic(5) (3 copies)
    1
> N := pCore(G,5);
> M := GModule(G,N);
> CM := CohomologyModule(G,M);

> G:=CyclicGroup(4);
> mats := [ Matrix(Integers(),2,2,[1,2,1,3]) ];
> invar := [2,4];
> CM := CohomologyModule(G,invar,mats);
> CM;
Cohomology Module

> G := Group<x,y | x^2,y^3,(x*y)^7 >;
> L := LowIndexSubgroups(G, <7,7>);
> L := LowIndexSubgroups(G, <7,7>);
> Index(G,L[1]);
7
> Q := CosetImage(G,L[1]);
> PM := PermutationModule(Q, Integers());
> cons := Constituents(PM);
> cons;
[
    GModule of dimension 1 over Integer Ring,
    GModule of dimension 6 over Integer Ring
]
> mats := ActionGenerators(cons[2]);
> M := GModule(G,mats);
> CM := CohomologyModule(G,M);