Exercise 6, § 6

Let $G$ un groupe fini d’ordre $n$, $M$ un $\mathbf{Z}^{(G)}$-module. Now prove that for all $q \geqslant 1$, the $\mathbf{Z}$-modules $H^q(G, M)$ and $H_q(G, M)$ are annihilated by $n$ (consider the homotopy $h$ of $C(G, M)$ into itself defined by

for $f \in C^{q+1}(G, M)$ and $g, g_1, ..., g_q \in G$. If $M$ is a finitely generated $\mathbf{Z}$-module, the groups $H^q(G, M)$ and $H_q(G, M)$ are finite for $q \geqslant 1$.