If |G| = p^2 q for two distinct primes p, q, then is G necessarily abelian? Are there any non-abelian groups of order p^2 q?
of course! for example $\displaystyle A_4$ is non-abelian and its order is $\displaystyle 12=2^2 \times 3.$ another example is the dihedral group of order 12. however, if G has order $\displaystyle p^2q, \ p <q,$ and $\displaystyle p \nmid q-1,$ then $\displaystyle G$ is abelian.