Nonabelian group of order pq
= Nonabelian group of order pq
Let $p>q$ be <prime numbers>. A nonabelian group of order $pq$ has a unique Sylow $p$-subgroup and exactly $p$ Sylow $q$-subgroups. Hence the Sylow count satisfies $p\equiv1\pmod q$, so $q\mid p-1$.