Exercise 7, § 6

Let $G$ be a cyclic group of order $n$, $M$ a $\mathbf Z^{(G)}$-module, $\sigma$ a generator of $G$. We denote by $N$ the multiplication in $M$ by the element $\displaystyle\sum_{g\in G}e_g$ of $\mathbf Z^{(G)}$. Now prove that the $\mathbf Z$-module $H^0(G,M)$ (resp. $H_0(G,M)$) is isomorphic to $\operatorname{Ker}(1-\sigma_M)$ (resp. $\operatorname{Coker}(1-\sigma_M)$); for odd $q$, the groups $H^q(G,M)$ and $H_{q+1}(G,M)$ are isomorphic to $\operatorname{Ker}N/\operatorname{Im}(1-\sigma_M)$; for even $q\geq 2$, $H^q(G,M)$ and $H_{q-1}(G,M)$ are isomorphic to $\operatorname{Ker}(1-\sigma_M)/\operatorname{Im}N$. (Use exercise 1, p. 178.)