Let be a group of order
Then, may or may not be simple.
For example, is an example of a non-simple group of order This can be easily verified by considering the subgroup This is clearly normal as is abelian.
On the other hand, consider the alternating group of degree It is known that is simple for Thus, is simple and moreover, it has order
Interestingly, it can be shown that is the only simple group (up to isomorphism) of order