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Archiv der Mathematik

, Volume 102, Issue 2, pp 101–108 | Cite as

On the number of conjugacy classes of π-elements in finite groups

  • Attila Maróti
  • Hung Ngoc Nguyen
Article

Abstract

Let G be a finite group and π be a set of primes. Put \({d_{\pi}(G) = k_{\pi}(G)/|G|_{\pi}}\), where \({k_{\pi}(G)}\) is the number of conjugacy classes of π-elements in G and |G| π is the π-part of the order of G. In this paper we initiate the study of this invariant by showing that if \({d_{\pi}(G) > 5/8}\) then G possesses an abelian Hall π-subgroup, all Hall π-subgroups of G are conjugate, and every π-subgroup of G lies in some Hall π-subgroup of G. Furthermore, we have \({d_{\pi}(G) = 1}\) or \({d_{\pi}(G) = 2/3}\). This extends and generalizes a result of W. H. Gustafson.

Mathematics Subject Classification (2000)

Primary 20E45 

Keywords

Finite groups Conjugacy classes π-elements 

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Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Alfréd Rényi Institute of MathematicsBudapestHungary
  2. 2.Fachbereich MathematikTechnische Universität KaiserslauternKaiserslauternGermany
  3. 3.Department of MathematicsThe University of AkronAkronUSA

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