Can $G$ of order $pqr$ be simple if it's generated by elements of orders $p,q$? Let $G$ be a group of order $pqr$, where $p$, $q$ and $r$ be three distinct primes. By Cauchy's theorem there exist three elements, $a$, $b$ and $c$, whose orders are $p$, $q$ and $r$, respectively. If the subgroup generated by $a$ and $b$ is the whole group, then I wonder if it is possible that there exists a proper normal subgroup of $G$.

More generally, let $G$ be a finite group with order $p_1p_2\cdots p_n$, where $p_1< p_2< \cdots < p_n$ are distinct primes.  Then by the normalizer-centralizer theorem, the Sylow $p_1$-subgroup is central in its normalizer, so there is a normal $p_1$-complement in $G$ by Burnside's transfer theorem.  Inductively, we can then show the Sylow $p_n$-subgroup is normal in $G$.
