Exercise 21, § 6

Soient $p$ et $q$ two prime numbers such that $p > q$. (a) Now prove that every finite group G of order $pq$ is an extension of $\mathbf{Z}/q\mathbf{Z}$ by $\mathbf{Z}/p\mathbf{Z}$ (notice that every $p$-Sylow subgroup of G is distinguished). If moreover $p \not\equiv 1 \pmod{q}$, prove that G is cyclic. (b) Deduce that the center of a group cannot be of index 69. (c) Now prove that, if $p \equiv 1 \pmod{q}$ there exists a group of order $pq$ whose center is reduced to $e$. (d) Now prove that every noncommutative group of order 6 is isomorphic to $\mathfrak{S}_3$.