On solubility of groups with finitely many centralizers

Abstract

For any group G, let C(G) denote the set of centralizers of G.<br />We say that a group G has n centralizers (G is a Cn-group) if |C(G)| = n.<br />In this note, we prove that every finite Cn-group with n ≤ 21 is soluble and<br />this estimate is sharp. Moreover, we prove that every finite Cn-group with<br />|G| < 30n+15<br />19 is non-nilpotent soluble. This result gives a partial answer to a<br />conjecture raised by A. Ashrafi in 2000