Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 1 / 10G / c / Solution

Interchange the two primes if necessary so that $p<q$. By the Sylow theorems, the number $n_q$ of Sylow $q$-subgroups satisfiesIf $n_q\ne 1$, then $n_q$ is $p$ or $p^2$. The first is impossible because $1<p<q$. The second would implySince $q>p$ is prime, it cannot divide $p-1$, so it must divide $p+1$. But $q>p$ then forces $q=p+1$, impossible because $p$ and $q$ are both odd. Therefore $n_q=1$. The unique Sylow $q$-subgroup is a nontrivial proper normal subgroup, and hence