Exercise 5, § 7

Let $G$, $F$ be two groups. a) For any extension $\mathscr{E} : F \to E \to G$ of G by F (I, p. 62), the operation of E on F by inner automorphisms defines a homomorphism $E \to \mathrm{Aut}(F)$, whence by passing to the quotient a group homomorphism $\theta : G \to \mathrm{Aut}(F)/\mathrm{Int}(F)$, called the homomorphism associated with the extension $\mathscr{E}$. Two isomorphic extensions of G by F have the same associated homomorphism. b) Conversely, consider a group homomorphism $\theta : G \to \mathrm{Aut}(F)/\mathrm{Int}(F)$. Let

be the mapping of passing to the quotient. Choose a mapping $\sigma$ of G into $\mathrm{Aut}(F)$ such that $p \circ \sigma = \theta$, and a mapping $f$ of $G \times G$ into F such that $\sigma(x) \sigma(y) (\sigma(xy))^{-1} = \mathrm{Int}(f(x, y))$ for $x, y \in G$; put $c(x, y, z) = \sigma(x)(f(y, z)).f(x, yz).(f(xy, z))^{-1}.(f(x, y))^{-1}$. Now prove that $c(x, y, z)$ belongs to the center Z of F. c) Consider Z as a G-module by putting $g.z = \sigma(g)(z)$ for $g \in G, z \in Z$; this definition is independent of the choice of $\sigma$. The mapping c defines an element of $C^3(G, Z)$; now prove that c is a 3-cocycle, and that its class in $H^3(G, Z)$ is independent of the choices of $\sigma$ and $f$. We denote it by $\omega(G, F, \theta)$. d) Now prove that the nullity of $\omega(G, F, \theta)$ is a necessary and sufficient condition for there to exist an extension of G by F having $\theta$ as associated homomorphism (if $\omega(G, F, \theta) = 0$, show that one can choose $f$ in such a way that the cocycle c is zero, then define a group law on $G \times F$). e) Suppose that $\omega(G, F, \theta) = 0$. Now prove that the set of isomorphism classes of extensions of G by F, with associated homomorphism $\theta$, is a principal homogeneous set under the group $H^2(G, Z)$. f) Let H be a group, M a $\mathbf{Z}^{(H)}$-module, x an element of $H^3(H, M)$. Now prove that there exists a group E with center M and a homomorphism $\varphi : H \to \mathrm{Aut}(E)/\mathrm{Int}(E)$, inducing on M the given H-module structure, such that $\omega(H, E, \varphi) = x$ (take $E = M \times L$, where L is the free group constructed on $H \times H$).