Proving $G$ is Abelian?

Question

Let $G$ be agroup of order $p^{2}q^{2}$, where $p$ and $q$ are distinct primes and $q \nmid p^{2} - 1$ and $p \nmid q^{2}-1$

How to prove that $G$ is Abelian?

Answer 1

We have a group of order $p^2q^2$ such that $q\nmid p^2 -1 $ and $p\nmid q^2 -1$ . By Sylow theorem $n_p\text{(Sylow p-subgroup)}=1+kp|q^2$ then $n_p = 1$,similarly $n_q\text{(Sylow q-subgroup)}=1$.So both will be normal in $G$.

Now let $H=\text{sylow $p $ subgroup and $K=$ sylow $q $ subgroup}$.$O(HK)=\frac{O(H)O(K)}{O(H\cap K)}=\frac{p^2q^2}{1}=p^2q^2$ i.e $G=H\times K$.So,$G$ is abelian(because $G$ is internal direct product of $H$ and $K$ and both are abelian).