# Math Help - Symmetric groups

1. ## Symmetric groups

Help me to prove this,

Show that if and with then .

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

3. 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?