Exercise 27, § 6

Let $S$ be a finite group. We say that $S$ is a minimal simple group if $S$ is simple, noncommutative, and if every subgroup of $S$ distinct from $S$ is solvable. a) Now prove that the alternating group $\mathfrak{A}_n$ is minimal simple if and only if $n = 5$. b) Let $G$ be a finite group. Now prove that, if $G$ is not solvable, there exist two subgroups $H$ and $K$ of $G$, with $H$ distinguished in $K$, such that $K/H$ is a minimal simple group.