Exercise 16, § 5

(a) Show that the alternating group $\mathfrak{A}_n$ is $(n-2)$-ply transitive.

(b) Show that $\mathfrak{A}_n$ is a non-commutative simple group for $n \geq 5$. (Use (a), the method of Exercise 15 and Exercise 12(d) to prove that $\mathfrak{A}_n$ is simple if $n > 6$; examine analogously the cases $n = 5$ and $n = 6$.)

(c) Show that, for $n \geq 5$, the only normal subgroups of $\mathfrak{S}_n$ are $\{e\}$, $\mathfrak{A}_n$ and $\mathfrak{S}_n$.