Exercise 21, § 6

Let $p$ and $q$ be two prime numbers such that $p > q$. (a) Show that every finite group $G$ of order $pq$ is an extension of $\mathbf{Z}/q\mathbf{Z}$ by $\mathbf{Z}/p\mathbf{Z}$ (note that every Sylow $p$-subgroup of $G$ is normal). If further $p \not\equiv 1 \pmod{q}$, show that $G$ is cyclic. (b) Deduce that the centre of a group cannot be of index 69. (c) Show that, if $p \equiv 1 \pmod{q}$, there exist a group of order $pq$ whose centre is $\{e\}$. (d) Show that every non-commutative group of order 6 is isomorphic to $\mathfrak{S}_3$.