The functions described in this section merely return data used to define the cohomology module. In each case, the argument (CM) must be a cohomology module returned by a call to CohomologyModule.
The K[G]-module used to define the cohomology module (CM). An error occurs if (CM) was defined by an action on a finitely generated abelian group.
Given a cohomology module (CM) that was defined by an action on a finitely generated abelian group A, return the invariants of A. If (CM) was not defined by an action on an abelian group, an error results.
Let (CM) be a cohomology module. If (CM) was defined by the action of a group on an R-module M, return the dimension of M. In the case in which (CM) was defined by the action of a group on a finitely generated abelian group A, the rank of A is returned.
The ring over which the module used to define the cohomology module (CM) is defined. If (CM) is defined in terms of an action on a finitely generated abelian group A, then the ring will be the integers if A is infinite, and the integers modulo the exponent of A if A is finite.
The group used to define action on the cohomology module (CM).
Given a cohomology module (CM) with associated group G, return a finitely presented group F isomorphic to G and the isomorphism from F to G. This presentation is on a strong generating set if G is a permutation or matrix group. It is used in the construction of presentations of extensions returned by the function Extension.
The matrix representing the action of the element g in the group of (CM) on the module of (CM).
> G:=CyclicGroup(4);
> mats := [ Matrix(Integers(),2,2,[1,2,1,3]) ];
> invar := [2,4];
> CM := CohomologyModule(G,invar,mats);
> Invariants(CM);
[ 2, 4 ]
> FPGroup(CM);
Finitely presented group on 1 generator
Relations
$.1^4 = Id($)
Mapping from: GrpFP to GrpPerm: G
> MatrixOfElement(CM, G.1^2);
[1 0]
[0 3]