For which n is there exactly one abelian group of order n?
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?