Abstract

In [11] and [12] Shahryari and Shahabi investigated the structure of finite groups containing a normal subgroup which is a union of two or three conjugacy classes. In [13] Riese and Shahabi investigated the similar problem for normal subgroups which are a union of four conjugacy classes. In [2], we investigated the structure of finite non-perfect groups in which every non-trivial proper normal subgroup is a union of n conjugacy classes, for a given integer n.

In this survey paper we report these results and investigate some new problems.

2000 Mathematics Subject Classification: 20E34, 20D10.

Keywords and phrases: Conjugacy class, normal subgroup

Introduction

Let G be a finite group and h be a non-central element of G. Following Shahryari and Shahabi [11], we say that a normal subgroup H of the group G is a small subgroup if H = 1 ∪ ClG(h), in which ClG(h) denotes the G-conjugacy class containing h. It is easy to see that H ≤ G′ and |H|(|H| − 1) ||G|. Moreover, H is an elementary abelian normal subgroup of G.

In [11], Shahryari and Shahabi studied the structure of finite centerless groups in which G′, the derived subgroup of G, is a small subgroup. They proved that:

Theorem 1.1 (Shahryari and Shahabi [11]) Let G be a finite centerless group and G′ be a small subgroup of G. Then:

1. (a) G is a Frobenius group with kernel G′ and its kernel is abelian.

2. (b) G has exactly one irreducible non-linear character χ.

3. […]