Originally Posted by

**skyking** I am having some difficulties here and would like some guidance.

Let $\displaystyle G$ be a group such that $\displaystyle |G|=p^2q^2$ where $\displaystyle p,q$ are primes and $\displaystyle p>q$ and $\displaystyle q$ is not a divisor of $\displaystyle p\pm 1$. Show that $\displaystyle G$ is abelian.

What I can determine so far is that $\displaystyle G$ has p-sylow group of order $\displaystyle p^2$ and a q-sylow group of order $\displaystyle q^2$ and that both must be abelian.

Additionally I can try to eliminate the possibilities for $\displaystyle s_p,s_q$ (the number of p,q-sylow groups).

What I got is that the options for $\displaystyle s_q$ are $\displaystyle 1, pq$ and for $\displaystyle s_p$ they are $\displaystyle 1,q^2$. but I can't get anywhere from here.

Any help appreciated.