Let $p$ and $q$ be integer primes such that $p$ divides $q-1$.
(a) Show that there exists a group $G$ of order $p^{2}q$ with generators $x$ and $y$ such that $x^{p^{2}} =1$, $y^{q}=1$, and $xyx^{-1}=y^{a}$, with $1$ the identity element of $G$ and $a$ some integer such that $a\not\equiv 1 \pmod q$ but $a^{p}\equiv 1\pmod q$.
(b) Prove that Sylow $q$-subgroup $S_{q}$ is normal in $G$, $G/S_{q}$ is cyclic and deduce that $G$ has a unique subgroup $H$ of order $pq$.