# Math Help - Index of Permutation groups

1. ## Index of Permutation groups

Let H= {phi element of Sn | phi(n)=n}
What's the index of H in Sn?

the index of H in Sn = |Sn|/|H|
and I know |Sn|= n!

I was told the order of H is (n-1)! but i don't see how
Could anyone help?

2. Originally Posted by frankdent1
Let H= {phi element of Sn | phi(n)=n}
What's the index of H in Sn?

the index of H in Sn = |Sn|/|H|
and I know |Sn|= n!

I was told the order of H is (n-1)! but i don't see how
Could anyone help?
Basically $H$ are all permutations that fix $n$. Therefore $H$ can be regarded as the group of all permutations of $\{ 1,2,...n-1\}$ i.e. isomorphic to $S_{n-1}$ and so $|H| = (n-1)!$.