Exercise 7, § 6

Let G and A be two groups, A being commutative. Let $\tau$ be a homomorphism of G into the automorphism group Aut(A) of A; if $g \in G, a \in A$, we write $^g a = \tau(g)(a)$. A crossed homomorphism of G into A is any mapping $\phi : G \to A$ such that

the group A being written additively. The crossed homomorphisms of G into A form a group under addition denoted by $Z(G, A)$.

(a) If $a \in A$, the mapping $g \mapsto ^g a - a$ is denoted by $\theta_a$. Show that $a \mapsto \theta_a$ is a homomorphism $\theta$ of A into $Z(G, A)$. The kernel of $\theta$ is the subgroup of A consisting of the elements invariant under G. The image of $\theta$ is denoted by $B(G, A)$.

(b) Let $X = A \times_{\tau} G$ be the semi-direct product of G by A. If $\phi$ is a mapping of G into A, show that $g \mapsto (\phi(g), g)$ is a section of X if and only if $\phi$ belongs to $Z(G, A)$. For the sections corresponding to $\phi_1, \phi_2 \in Z(G, A)$ to be conjugate by an element of A, it is necessary and sufficient that $\phi_1 \equiv \phi_2 \mod B(G, A)$.

(c) Let $F \xrightarrow{i} E \xrightarrow{p} G$ be an extension of $G$ by a group $F$ whose centre is equal to $A$. Suppose that, if $y \in E$ and $x = p(y)$, then

for all $f \in A$. If $\phi \in Z(G, A)$, let $u_\phi$ be the mapping of $E$ into $E$ given by the formula $u_\phi(y) = i(\phi(p(y))) . y$. Show that $\phi \mapsto u_\phi$ is an isomorphism of $Z(G, A)$ onto the automorphism group $\mathrm{Aut}(E)$ of the extension $E$. This isomorphism maps $B(G, A)$ to the group of automorphisms $\mathrm{Int}_E(x)$, where $x$ runs through $A$.