# Unique abelian group of order n

#### hmw

1. Homework Statement

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

2. Homework Equations

3. The Attempt at a Solution

All prime integers?

Related Calculus and Beyond Homework Help News on Phys.org

#### Dick

Homework Helper
Shouldn't you prove it?

#### 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?

### Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving