## A Group of Order the Square of a Prime is Abelian

## Problem 20

Suppose the order of a group $G$ is $p^2$, where $p$ is a prime number.

Show that

**(a)** the group $G$ is an abelian group, and

**(b) **the group $G$ is isomorphic to either $\Zmod{p^2}$ or $\Zmod{p} \times \Zmod{p}$ without using the fundamental theorem of abelian groups.