TITLE: Give examples of two non-isomorphic groups of order $n^2$
QUESTION [5 upvotes]: For each integer, $n > 7$, give examples of two non-isomorphic groups of order $n^2$.

REPLY [4 votes]: How about $\mathbb{Z}/n^{2}\mathbb{Z}$ and $\mathbb{Z}/n\mathbb{Z}\oplus \mathbb{Z}/n\mathbb{Z}$. One is cyclic and the other not.

REPLY [3 votes]: Hint: Think of $\mathbb{Z}_n$ and direct product of this.