# Unique abelian group of order n

1. Apr 19, 2008

### hmw

1. The problem statement, all variables and given/known data

Determine all integers for which there exists a unique abelian group of order n.

2. Relevant equations

3. The attempt at a solution

All prime integers?

2. Apr 19, 2008

### Dick

Shouldn't you prove it?

3. Mar 17, 2011

### The Chaz

Since this was the first hit when I Googled my homework, I'll resurrect this thread with my thoughts.
This holds trivially for prime numbers.
Claim: For n "square-free", $$\mathbb{Z}_n$$ is (up to isomorphism) the unique abelian group of order n.

We extend the fact that $$\mathbb{Z}_{ab} \cong \mathbb{Z}_a \times \mathbb{Z}_b \Leftrightarrow gcd(a, b) = 1$$ by induction to an arbitrary amount of factors. Then the factors (which are also the subscripts of Z, and the orders of the groups) are pairwise coprime, and we have our result.

Thoughts?