Exercise 1, § 6

Let G be a group and H a normal subgroup of G such that G/H is cyclic of finite order n. Let x be an element of G whose image $\bar{x}$ in G/H generates G/H. Let $\phi$ be the automorphism $h \mapsto xhx^{-1}$ of H and let $y = x^n$; then $y \in H$. Show that $\phi(y) = y$ and that $\phi^n$ is the inner automorphism of H defined by y.

Let $\tau$ be the operation of $\mathbf{Z}$ on H defined by $(m, h) \mapsto \phi^m(h)$ and let $E = H \times_{\tau} \mathbf{Z}$ be the corresponding semi-direct product. Show that the element $(y^{-1}, n)$ of E generates a central subgroup C: of E and that the quotient $E/C_y$ is isomorphic to G.