On solubility of groups with finitely many centralizers

Document Type: Research Paper

Author

University of Kurdistan

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

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