# Math Help - prove that Sn is the maximal subgroup of Sn+1.

1. ## prove that Sn is the maximal subgroup of Sn+1.

Just like the title says, this is probably quite elementary but I couldn't quite get started.

2. Originally Posted by ChiliNBeans
Just like the title says, this is probably quite elementary but I couldn't quite get started.

Not "the" but "one" maximal subgroup, since there are others, say the alternating subgroup $A_{n+1}$.

Now, $S_n$ as subset of $S_{n+1}$ can be identified as the set (in fact subgroup) of all permutations which leave one number

out of $\{1,2,...,n\}$ fixed, so if $S_n< N\leq S_{n+1}$ , then N must move all the numbers in $\{1,2,...,n\}$ . Try to use this to show that then $N=S_{n+1}$

Tonio