This problem is not homework but, I was stuck to it when I reviewed the Sylow theorems and problems. I am really interested of finding a test in which we can examine whether a finite group of certain order is abelian or not. It tells:
$G$ is a finite group of order $p^2q$ wherein $p$ and $q$ are distinct primes such that $p^2 \not\equiv 1$ (mod $q$) and $q \not\equiv 1$ (mod $p$). Then $G$ is an abelian group.
We know that $n_p=1+kp$ and it must divide $p^2q$. So, $1+kp|q$ and because of $q≢1$ (mod $p$), we get $n_p=1$. This means that we have a unique $p$-sylow of $G$, for example $P$, in the group and so is normal and ofcourse isomorphic to $\mathbb Z_{p^2}$ or $\mathbb Z_p×Z_p$. What should I do next? Thanks for helping me.