Exercise 4, § 6

For every group $G$, we denote by $\varepsilon_G : \mathbf{Z}^{(G)} \to \mathbf{Z}$ the ring homomorphism such that $\varepsilon_G(e_g) = 1$ for all $g \in G$; we set $I_G = \operatorname{Ker}(\varepsilon_G)$. a) Prove that the group $H_1(G, \mathbf{Z})$ is isomorphic to $I_G/I_G^2$. b) Let $(g_i)_{i \in I}$ be a generating family of $G$; prove that the elements $(e_{g_i} - 1)$, for $i \in I$, generate the ideal $I_G$. c) Suppose that the family $(g_i)_{i \in I}$ is basic (I, p. 86). Prove that $(e_{g_i} - 1)_{i \in I}$ is a basis of the $\mathbf{Z}^{(G)}$-left module $I_G$.