# Thread: Non-abelian groups of order p^2 q

1. ## Non-abelian groups of order p^2 q

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?

2. Originally Posted by prettyboy
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 $A_4$ is non-abelian and its order is $12=2^2 \times 3.$ another example is the dihedral group of order 12. however, if G has order $p^2q, \ p and $p \nmid q-1,$ then $G$ is abelian.

3. ## refer book

Refer the book ion abstract algebra by Dummit and Foote. a large behaviour of these groups are well explained