Universal coefficient theorem for group homology
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:
Then we have:
where we have:
If, further, is a finitely generated abelian group, of the form:
Then the expressions simpliy further: