Simplicity of normal subgroups and conjugacy class sizes

Abstract

Given a finite group \(G\) which possesses a non-abelian simple normal subgroup \(N\) having exactly four \(G\)-class sizes, we prove that \(N\) is isomorphic to PSL\((2, 2^a)\) with \(a\ge 2\). Thus, we obtain an extension for normal subgroups of the classic N. Ito’s theorem which characterizes those finite simple groups with exactly four class sizes.