Exercise 27, § 6

Let S be a finite group. S is called a minimal simple group if S is simple and non-commutative and if every subgroup of S distinct from S is solvable.

(a) Show that the alternating group $\mathfrak{A}_n$ is minimal simple if and only if $n = 5$.

(b) Let G be a finite group. Show that, if G is not solvable, there exist two subgroups H and K of G with H normal in K such that K/H is a minimal simple group.