If $G$ is a non-cyclic group of order $n^2$, then $G$ is isomorphic to $\mathbb{Z_n} \oplus \mathbb{Z_n}$ I've independently come up with a question (I know it's been asked before, but I can't find the question online) involving the external direct product, non-cyclic groups and isomorphisms.  So, is the following statement true?  
Claim:  "If $G$ is a non-cyclic group of order $n^2$, then $G$ is isomorphic to $\mathbb{Z_n} \oplus \mathbb{Z_n}$."
Where $\mathbb{Z_n} \oplus \mathbb{Z_n}$ is the external direct product of $\mathbb{Z_n}$ and $\mathbb{Z_n}$.
I've been thinking about this for a few hours, but really can't figure out a good place to start (or a counter-example).  

The question should be formulated as follows.


*

*Let $p$ be a prime.  Consider an abelian group $A$ of order $p^2.$  Prove that either $A$ is cyclic or $A={\mathbb Z}/p{\mathbb Z}\oplus {\mathbb Z}/p{\mathbb Z}.$

*Let $n$ have three or more divisors (i.e. either there are two different primes dividing $n$, or there is one prime $p$ such that $p^2|n$).  Then there are more than two abelian groups of order $n^2$, up to isomorphism, that are non cyclic.
I would suggest the following to start with: think of Lagrange's theorem (for the first part).
