On solubility of groups with finitely many centralizers

2012/05/03 by Mohammad Zarrin, Zarrin, Mohammad
Engineering · Mathematics · #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Rings, Modules, and Algebras #graph theory and CDMA systems #math.GR

paper · pdf · doi:10.48550/arxiv.1205.0643

to appear in Bulletin of the Iranian Mathematical Society

arxiv created 2012/05/03 · openalex publication_date 2012/05/03 · arxiv updated 2012/05/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

• On finite groups with soluble centralisers
• Centralisers of Finite Subgroups in Soluble Groups of Type FP_n
• On a problem of B. Hartley about a small centralizer in finite and locally finite groups
• Centralizer fusion systems of central involutions in a finite group with soluble centralizer of involutions
• Profinite groups with soluble centralisers