Exercise 25, § 5

Let $n$ be an integer $\geq 5$. Show that the subgroups of $\mathfrak{S}_n$ of order $(n-1)!$ are isomorphic to $\mathfrak{S}_{n-1}$ and form a single conjugacy class (resp. two conjugacy classes) if $n \neq 6$ (resp. if $n = 6$). (Let $H$ be such a subgroup and $x_1, \ldots, x_n$ the elements of $\mathfrak{S}_n / H$. Show that the action of $\mathfrak{S}_n$ on the $x_i$ defines an automorphism of $\mathfrak{S}_n$ and apply Exercises 23 and 24.)