On solubility of groups with finitely many centralizers

Document Type : Research Paper


University of Kurdistan


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+15
19 is non-nilpotent soluble. This result gives a partial answer to a
conjecture raised by A. Ashrafi in 2000


Main Subjects

  • Receive Date: 25 December 2011
  • Revise Date: 29 April 2012
  • Accept Date: 30 April 2012
  • First Publish Date: 01 July 2013