The group of cohomology $H^3(G,\mathbb{Z})$ is finite when $G$ is finite.

Question

The group of cohomology $H^3(G,\mathbb{Z})$ is finite when G is finite.

I am not sure how this is finite. We use the definite as follows:

$H^n(G,K) = Ext_\mathbb{Z}^n$$_G (\mathbb{Z}, K)$ and we use the $G$-free resolution of $\mathbb{Z}$.

Any help would be appreciated!

Answer 1

When $G$ is a finite group of order $n$, then $H^k(G,A)$ is $n$-torsion for all $k\ge1$.

From the standard resolution, $H^k(G,A)$ is finitely generated whenever $A$ is finitely generated as an Abelian group. As $H^k(G,A)$ is both finitely generated and torsion, it is a finite Abelian group.