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.