# Universal coefficient theorem for group homology

From Groupprops

## Contents

## Statement

### For coefficients in an abelian group

Suppose is a group and is an abelian group. The **universal coefficients theorem for group homology** describes the homology groups for trivial group action of on in terms of the homology groups for trivial group action of on .

Explicitly, it states that there is a natural short exact sequence of abelian groups:

The sequence splits (though not naturally) to give that:

### Typical case of finitely generated abelian groups

Suppose for some finite group and for some finite group .

Suppose further that:

and

Then we have:

where we have:

and

where

Thus, overall:

If, further, is a finitely generated abelian group, of the form:

Then the expressions simpliy further:

and