On solubility of groups with finitely many centralizers

Abstract. For any group G, let C(G) denote the set of centralizers of G. We say that a group G has n centralizers (G is a Cn-group) if |C(G) | = n. In this note, we prove that every finite Cn-group with n ≤ 21 is soluble and this estimate is sharp. Moreover, we prove that every finite Cn-group with |G | < 30n+1519 is non-nilpotent soluble. This result gives a partial answer to a conjecture raised by A. Ashrafi in 2000. 1