Exercise 3, § 7

Let $G$ be a group, $A$ the algebra $\mathbf{Z}^{(G)}$, $j:G\to A$ the canonical injection. We endow $A$ with the augmentation $\pi:A\to\mathbf{Z}$ defined by $\pi(e_g)=1$ for all $g\in G$. Let $M$ be a left $A$-module.

a) Let $0\to M_{(\pi)}\to B\xrightarrow{p}A\to0$ be an extension of $A$ by the $(A,A)$-bimodule $M_{(\pi)}$ (X, p. 196, exercise 27). We denote by $\Gamma$ the subset of $B$ formed by the elements $b$ of $B$ such that $p(b)\in j(G)$. Show that $\Gamma$, endowed with the multiplication induced by that of $B$, is a group, extension of $G$ by the $G$-module $M$.

b) The preceding construction defines a mapping from $Ex_{\mathbf Z}(A,M_{(\pi)})$ into $Ex(G,M)$ (VIII, § 13). Show that the diagram

is commutative.