Example 4

Let $M$ be a $\mathbf{Z}$-module, considered as a right $\mathbf{Z}^{(G)}$-module by making $G$ operate trivially. The group $H_1(G,M)$ is isomorphic to the quotient of $M\otimes_{\mathbf{Z}}\mathbf{Z}^{(G)}$ by the sub-$\mathbf{Z}$-module generated by the elements $m\otimes(e_{g_1g_2}-e_{g_1}-e_{g_2})$ for $m$ in $M$, $g_1,g_2$ in $G$; it follows easily that $H_1(G,M)$ is isomorphic to $M\otimes_{\mathbf{Z}}(G/(G,G))$.

Let us denote by $\sigma$ the anti-automorphism of $\mathbf{Z}^{(G)}$ defined by $\sigma(e_g) = e_{g^{-1}}$ for $g \in G$. Every left $\mathbf{Z}^{(G)}$-module may be considered as a right $\mathbf{Z}^{(G)}$-module by means of $\sigma$, and conversely. This makes it possible, for example, to define the groups $H_q(G, M)$ for a left $\mathbf{Z}^{(G)}$-module $M$, by setting $H_q(G, M) = H_q(G, \sigma_*(M)) = \mathrm{Tor}_q^{\mathbf{Z}^{(G)}}(\mathbf{Z}, M)$.