Normal sections, class sizes and solvability of finite groups

Abstract

If G is a finite group, we show that any normal subgroup of G which has exactly three G-conjugacy class sizes is solvable. Thus, we give an extension for normal subgroups of the classical N. Itô’s theorem which asserts that those finite groups having three class sizes are solvable, and particularly, a new proof of it is provided. In order to do this, we investigate the structure of a normal section N/K of G such that every element in N lying outside of K has the same G-class size.