Show there is no non-abelian group of order 9

Question

I want to show there is no non-abelian group of order 9. How should I attempt this?

Answer 1

The result is true for any order that is the square of a prime.

The center of a group of order $p^2$ cannot be trivial because all of the conjugacy classes that are not in the center have order multiple of $p$ ( This is true for any group of order $p^\alpha$)

Therefore $\frac{G}{Z(G)}$ is cyclic, and if this happens $G$ is abelian.

Why?

let $a$ be a generator of $\frac{G}{Z(G)}$. Then every element in $G$ is of the form $a^nx$ with $x\in Z(G)$. take two elements in $G$: $a^nx_1$ and $a^mx_2$. We have:

$a^nx_1a^mx_2=a^na^mx_1x_2=a^na^mx_2x_1=a^ma^nx_2x_1=a^mx_2a^nx_1$. So every two elements of $G$ commute.