Exercise 8, § 6

With the notation of the above exercise, $C(G, A)$ is used to denote the group of mappings of $G$ into $A$. $G$ operates on $C(G, A)$ by the formula

If $\sigma$ denotes the corresponding homomorphism of $G$ into $\mathrm{Aut}(C(G, A))$, the semi-direct product $C(G, A) \times_\sigma G$ is denoted by $E_0$.

(a) Let $\varepsilon : A \to C(G, A)$ be the mapping which associates with each $a \in A$ the constant mapping equal to $a$. Show that $\varepsilon$ is an injective homomorphism, compatible with the action of $G$.

(b) Let $A \xrightarrow{i} E \xrightarrow{p} G$ be an extension of $G$ by $A$ such that $i(ga) = xi(a)x^{-1}$ if $a \in A, x \in E$ and $g = p(x)$. Let $\rho$ be a mapping $G \to E$ such that $p \circ \rho = \mathrm{Id}_G$. For all $x \in E$, let $\phi_x$ be the mapping of $G$ into $A$ such that

Show that Deduce that the mapping $\Phi : E \to E_0$ defined by is a homomorphism which makes the following diagram commutative:

(c) A $G$-mean on $A$ is a homomorphism

satisfying the two following conditions:

Let $m$ be a G-mean on A. Show that, if $\phi$ is a crossed homomorphism and $a = m(\phi)$, then $\phi = \theta_{-a}$ (cf. Exercise 7). In particular,

Show that the mapping $(\phi, g) \mapsto (m(\phi), g)$ is a homomorphism of $E_0$ into $A \times_{\tau} G$. Deduce that every extension E of G by A satisfying the condition in (b) is isomorphic to $A \times_{\tau} G$ (use the composition $E \xrightarrow{\Phi} E_0 \to A \times_{\tau} G$) and hence admits a section. Show that two such sections are transformed one into the other by an inner automorphism of E defined by an element of A (use Exercise 7 and the fact that $Z(G, A) = B(G, A)$).

(d) Suppose that G is finite of order n and that the mapping $\alpha \mapsto n\alpha$ is an automorphism of A. For all $\phi \in C(G, A)$, let $m(\phi)$ denote the unique element of A such that

Show that $m$ is a G-mean on A.

This applies in particular when A is finite of order prime to n.