# Accessing Properties of the Cohomology Module

The functions described in this section merely return data used to define the cohomology module. In each case, the argument ${\rm CM}$ must be a cohomology module returned by a call to `CohomologyModule`.

## `Module(CM): ModCoho -> ModGrp`

The $K[G]$-module used to define the cohomology module ${\rm CM}$. An error occurs if ${\rm CM}$ was defined by an action on a finitely generated abelian group.

## `Invariants(CM): ModCoho -> SeqEnum`

Given a cohomology module ${\rm CM}$ that was defined by an action on a finitely generated abelian group $A$, return the invariants of $A$. If ${\rm CM}$ was not defined by an action on an abelian group, an error results.

## `Dimension(CM): ModCoho -> RngIntElt`

Let ${\rm CM}$ be a cohomology module. If ${\rm CM}$ was defined by the action of a group on an $R$-module $M$, return the dimension of $M$. In the case in which ${\rm CM}$ was defined by the action of a group on a finitely generated abelian group $A$, the rank of $A$ is returned.

## `Ring(CM): ModCoho -> ModGrp`

The ring over which the module used to define the cohomology module ${\rm CM}$ is defined. If ${\rm CM}$ 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.

## `Group(CM): ModCoho -> Grp`

The group used to define action on the cohomology module ${\rm CM}$.

## `FPGroup(CM): ModCoho -> Grp, HomGrp`

Given a cohomology module ${\rm CM}$ 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`](extensions.md#function-grpcoh-extension).

## `MatrixOfElement(CM, g): ModCoho, GrpElt -> AlgMatElt`

The matrix representing the action of the element $g$ in the group of ${\rm CM}$ on the module of ${\rm CM}$.

## `Example: Coho Module2cont (ex-fdbfad)`

Following on from example [Example: Coho Module2](creation.md#example-ex-0c0fe6) above:

```magma
> 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]

```