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 .
Now, as subset of can be identified as the set (in fact subgroup) of all permutations which leave one number
out of fixed, so if , then N must move all the numbers in . Try to use this to show that then
Tonio