Exercise 29, § 2

Let $H$ be a simple non-abelian group. Let $\theta$ be an automorphism of $H$ of prime order $p$ and let $U$ be the semi-direct product of $\mathbf{Z}/p\mathbf{Z}$ by $H$ corresponding to $\theta$. Show that, if $\theta$ is not an inner automorphism, the only normal subgroups of $U$ are $\{1\}$, $H$ and $U$. Deduce that $U$ is not soluble, but that it satisfies condition (R) of no. 7. Apply this to show that the symmetric group $\mathfrak{S}_n$ satisfies condition (R).