Subgroup of Symmetric Group that Fixes n

Theorem
Let $S_n$ denote the symmetric group on $n$ letters.

Let $H$ denote the subgroup of $S_n$ which consists of all $\pi \in S_n$ such that:
 * $\map \pi n = n$

Then:
 * $H = S_{n - 1}$

and the index of $H$ in $S_n$ is given by:


 * $\index {S_n} H = n$