Question on groups of order $pq$
Let $G$ be a group of order $pq$, $p>q$ and $p$, $q$ are primes. Then prove that
- If $q\mid p-1$ then there exists a non abelian group of order $pq$.
- Any two non-abelian groups of order $pq$ are isomorphic.
I have proved that if $q\not\mid p-1$ then $G$ is cylic . But how to prove this one I have no idea. Any kind of hint is very much welcome. This problem is in Herstein book, page 75.
Solution 1:
If $q\mid p-1$ then $\rm{Aut}(C_p)$ has a unique subgroup of order $q$, and the map embedding $C_q$ in $\rm{Aut}(C_p)$ gives a semidirect product, which is not abelian (easy to check).
On the other hand, if $G$ is some other non-abelian group of order $pq$ then it is an easy exercise that $G$ has a normal subgroup of order $p$ and since $G$ also has a subgroup of order $q$, it must be a semidirect product. But since the subgroup of order $q$ in $\rm{Aut}(C_p)$ was unique, the only possibility is the one we already accounted for.