Let \(G\) be a finite group of order \(|G| = p^k m\), where \(p\) is a prime. Let \(n_p\) denote the number of Sylow \(p\)-subgroups of \(G\). Suppose that \(n_p\) divides an integer \(q < p\). Prove that \(n_p = 1\), and consequently every Sylow \(p\)-subgroup is normal in \(G\).
Solution
By Sylow’s theorem, \[ n_p\equiv1\pmod p. \] The hypothesis \(n_p\mid q\) with \(0<q<p\) gives \[ 1\le n_p\le q<p. \] The only positive integer less than \(p\) that is congruent to \(1\) modulo \(p\) is \(1\). Hence \[ \boxed{n_p=1}. \] Therefore the Sylow \(p\)-subgroup is unique, and hence normal in \(G\).