Introduction

In this survey G denotes a finite group of order g, with k conjugacy classes and center Z(G) of order z. Denote the order of G', the commutator subgroup of G, by g' and assume that g > 1. Denote by Cls(G) = {c1 = 1, c2,…,ck} the multiset composed of the sizes of the conjugacy classes of G (with c1 = |{1}|) and by Chd(G) = {x1 = 1, x2,…, xk} the multiset composed of the degrees of the irreducible characters of G (with x1 = 1G(1)). The influence of the arithmetical structure of the ci's and the xi's on the group-theoretical structure of G has been investigated in many papers. For example, the following results concerning the class sizes in G were proved in [4]. Here, and in the sequel, by “a class” we mean “a conjugacy class” and by “a prime” we mean “a non-necessarily fixed prime”. For additional information, see [4], [5], [6] and [8].

Theorem 1The following statements hold:

1. If ci equals 1 or a prime for each ci ∈ Cls(G), then either G is nilpotent of class ≤ 2 or G/Z(G) is a Frobenius group of order pq, where p and q are distinct primes.

2. If ci equals 1 or a prime power for each ci ∈ Cls(G), then either G is nilpotent or G/Z(G) is a solvable Frobenius group.

3. […]