
Symmetric groups

For each $\displaystyle r\in\{1,\ldots,n\}$, let $\displaystyle X_r$ be the set of all permutations in $\displaystyle S_n$ which fix r. Then $\displaystyle X_r$ is a set of permutations on n1 numbers, so $\displaystyle X_r=(n1)!$  hence $\displaystyle S_n:X_r=n$, and there is a onetoone correspondence between $\displaystyle X_r$ and $\displaystyle S_{n1}$

Here is a similar question I was thinking of.
Find all the index n subgroups of S_n.
Is it S(1), ... S(n)  where S(i) are permutations which fix i?
I know this is true for n=3,4.
But how do we prove it if it is true?