Finite groups with exactly three \(p\)-regular classes (Q1193245)

The following Theorem is proved: Let \(G\) be a finite nonsolvable group with \(O_ p(G)=1\). If \(G\) has exactly three \(p'\)-conjugacy classes, then either \(p=2\) and \(G \cong \Sigma_ 5\), \(PGL_ 2(7)\), \(M_{10}\), \(P \Gamma L_ 2(9)\) or \(p=3\) and \(G \cong P \Gamma L_ 2(8)\) or \(p=5\) and \(G \cong A_ 5\). The proof depends on the classification of finite simple groups.