On the number of centralizers and conjugacy class sizes in finite groups

Given a finite group G, denote by cs (G) the set of the sizes of the conjugacy classes of G and by Cent (G) the set of the centralizers of elements of G. Consider a prime p and integers s≥2 and n≥2, with gcd (p, s) =1. In this paper, some relations between cs (G) and |Cent (G) | are established in the case where cs (G) =1, pn, pn−1s. Further, when p∈2, 3, we determine the values of s and the structure of a finite group G such that cs (G) =1, pn, pn−1s. We also describe the structure of an AC-group G such that G: Z (G) =3ns and |Cent (G) |=1+∑i=0n3i, where gcd (3, s) =1.