For which

is there exactly one abelian group of order

?

I know that the answer is the product of distinct primes, but I need to say this in a formal proof.

I started out the proof doing this:

Suppose there is a group

such that

. The elementary divisors are uniquely determined by the order of the group if and only if

are distinct primes. So then G can be written uniquely in one way - the direct sum of its cyclic groups.

Thus, for

a product of distinct primes, there is one abelian group of order

.

Is there something I can do to make this answer stronger?

Thanks,

crushingyen