Let \(G\) be a group of order \(mp^n\) where \(p\) is a prime, \(m\) is a positive integer with \(1 < m < p.\) We show that \(G\) is not simple.
By Sylow Theorem (3), \(n_p \mid m\) and \(n_p \equiv 1 \mod p\).
This means that \(n_p = 1 + kp\) for some \(k \in \mathbb{Z}^+\cup\{0\}\). However, \(m < p\) forces \(k = 0\). Thus, \(n_p = 1\) and we are done.