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Journal of Group Theory

Editor-in-Chief: Parker, Christopher W. / Wilson, John S.

Managing Editor: Khukhro, Evgenii I. / Kramer, Linus

6 Issues per year


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Online
ISSN
1435-4446
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Volume 15, Issue 1 (Jan 2012)

Issues

Centralizers of subgroups in simple locally finite groups

Kıvanç Ersoy / Mahmut Kuzucuoğlu

Abstract

Hartley asked the following question: Is the centralizer of every finite subgroup in a simple non-linear locally finite group infinite? We answer a stronger version of this question for finite 𝒦-semisimple subgroups. Namely let G be a non-linear simple locally finite group which has a Kegel sequence 𝒦 = {(Gi, 1) : i ∈ ℕ} consisting of finite simple subgroups. Then for any finite subgroup F consisting of 𝒦-semisimple elements in G, the centralizer CG(F) has an infinite abelian subgroup A isomorphic to a direct product of ℤpi for infinitely many distinct primes pi.

Moreover we prove that if G is a non-linear simple locally finite group which has a Kegel sequence 𝒦 = {(Gi, 1) : i ∈ ℕ} consisting of finite simple subgroups Gi and F is a finite 𝒦-semisimple subgroup of G, then CG(F) involves an infinite simple non-linear locally finite group provided that the finite fields ki over which the simple group Gi is defined are splitting fields for Li, the inverse image of F in Ĝi for all i ∈ ℕ. The group Ĝi is the inverse image of Gi in the corresponding universal central extension group.

About the article

Received: 2009-10-16

Revised: 2010-12-13

Published in Print: 2012-01-01


Citation Information: Journal of Group Theory, ISSN (Online) 1435-4446, ISSN (Print) 1433-5883, DOI: https://doi.org/10.1515/JGT.2010.087.

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