Page "Simple group" Paragraph 25

Proof: If n is a prime-power, then a group of order n has a nontrivial center and, therefore, is not simple.

If n is not a prime power, then every Sylow subgroup is proper, and, by Sylow's Third Theorem, we know that the number of Sylow p-subgroups of a group of order n is equal to 1 modulo p and divides n. Since 1 is the only such number, the Sylow p-subgroup is unique, and therefore it is normal.

Since it is a proper, non-identity subgroup, the group is not simple.