Abstract

Let N be a normal subgroup of a finite group G. In the recent past years some results have appeared concerning the influence of the G-class sizes of N, that is, with the sizes of the conjugacy classes in G contained in N, on the structure of N. In this survey, we present the main results and techniques used for proving that any normal subgroup of G which has exactly three G-conjugacy class sizes is solvable. Thus, we obtain a generalisation for normal subgroups of the classical N. Itô's theorem which asserts that those finite groups having three class sizes are solvable, and in particular, a new proof of it is provided.

Introduction

The solvability of a finite group G with three conjugacy class sizes is a complex problem solved by N. Itô in [22]. He proved that such groups are solvable by appealing to Feit-Thompson's theorem and some deep classification theorems by M. Suzuki. This result was simplified by J. Rebmann in [25] when G is an F-group (that is, G has no pair of non-central elements such that the centraliser of one element properly contains the other centraliser). Then he determined the structure of F-groups by using results of R. Baer ([8] and [9]) and M. Suzuki ([27]) about groups with a non-trivial normal partition. Afterwards, A.R. Camina proved in [14], by using the description of finite groups with dihedral Sylow 2-subgroups given by D. Gorenstein and J.H. Walter, that if G is not an F-group and has three class sizes, then G is a direct product of an abelian subgroup and a subgroup whose order involves no more than two primes. Forty years later, the structure of these groups has been completely determined (up to nilpotent groups, which in this context are p-groups) by S. Dolfi and E. Jabara in [15], who based their proof on the solvability of this type of groups.