Show that Sylow \(p\)- and \(q\)-subgroups for distinct primes \(p\ne q\) intersect trivially.
Solution
Let \(P\) be a Sylow \(p\)-subgroup and \(Q\) a Sylow \(q\)-subgroup, with \(p\ne q\). Then \[ P\cap Q\le P\quad\text{and}\quad P\cap Q\le Q. \] Hence \(|P\cap Q|\) divides both a power of \(p\) and a power of \(q\). Since these powers are coprime, \(|P\cap Q|=1\). Thus \[ P\cap Q=1. \]