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(Pro)-finite and (topologically) locally finite groups with a CC-subgroup

Published online by Cambridge University Press:  11 January 2010

Z. Arad
Affiliation:
Department of Mathematics, Bar–Ilan University, Ramat–Gan, Israel and Department of Computer Science and Mathematics, Netanya Academic College, Netanya, Israel
W. Herfort
Affiliation:
Institut für Angewandte und Numerische Mathematik, Technische Universität Vienna, Austria; The second author would like to thank for greatful hospitality at the Bar-Ilan University, the Netanya Academic College and the Tel-Aviv University in February 2000
C. M. Campbell
Affiliation:
University of St Andrews, Scotland
E. F. Robertson
Affiliation:
University of St Andrews, Scotland
G. C. Smith
Affiliation:
University of Bath
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Summary

Abstract

A proper subgroup H of a group G is called a CC-subgroup of G if the centralizer CG(h) of hH# = H \ {1} is contained in H. Such finite groups were partially classified by G. Frobenius, W. Feit, K. W. Gruenberg and O. H. Kegel, J.S. Williams, A. S.Kondrat'iev, N. Iiyori and H.Yamaki, M. Suzuki, M. Herzog, Z. Arad, D. Chillag, Ch. Praeger and others.

In this report, using the classification of finite simple groups, we give a complete list of all finite groups containing a CC-subgroup. As a corollary we classify infinite profinite groups, locally finite groups and certain classes of topological groups containing a CC-subgroup under certain conditions.

Introduction

Let G denote a finite group. According to M. Herzog [18] a subgroup MG is a CC-subgroup (”centralizers contained“), if CG(m) ≤ M for every mM \ {1}. The example with smallest cardinality is G := S3 with either M := 〈(123)〉 or M := 〈(12)〉 being a CC-subgroup. More generally, by the well known result of G. Frobenius, every Frobenius group has CC-subgroups either the kernel or any complement.

Sketching the thread

One finds the concept of a CC-subgroup (without calling it that) in work of W. Feit describing doubly transitive groups which fix 3 letters (e.g. in [13]).

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Publisher: Cambridge University Press
Print publication year: 2003

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  • (Pro)-finite and (topologically) locally finite groups with a CC-subgroup
    • By Z. Arad, Department of Mathematics, Bar–Ilan University, Ramat–Gan, Israel and Department of Computer Science and Mathematics, Netanya Academic College, Netanya, Israel, W. Herfort, Institut für Angewandte und Numerische Mathematik, Technische Universität Vienna, Austria; The second author would like to thank for greatful hospitality at the Bar-Ilan University, the Netanya Academic College and the Tel-Aviv University in February 2000
  • Edited by C. M. Campbell, University of St Andrews, Scotland, E. F. Robertson, University of St Andrews, Scotland, G. C. Smith, University of Bath
  • Book: Groups St Andrews 2001 in Oxford
  • Online publication: 11 January 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511542770.004
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  • (Pro)-finite and (topologically) locally finite groups with a CC-subgroup
    • By Z. Arad, Department of Mathematics, Bar–Ilan University, Ramat–Gan, Israel and Department of Computer Science and Mathematics, Netanya Academic College, Netanya, Israel, W. Herfort, Institut für Angewandte und Numerische Mathematik, Technische Universität Vienna, Austria; The second author would like to thank for greatful hospitality at the Bar-Ilan University, the Netanya Academic College and the Tel-Aviv University in February 2000
  • Edited by C. M. Campbell, University of St Andrews, Scotland, E. F. Robertson, University of St Andrews, Scotland, G. C. Smith, University of Bath
  • Book: Groups St Andrews 2001 in Oxford
  • Online publication: 11 January 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511542770.004
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • (Pro)-finite and (topologically) locally finite groups with a CC-subgroup
    • By Z. Arad, Department of Mathematics, Bar–Ilan University, Ramat–Gan, Israel and Department of Computer Science and Mathematics, Netanya Academic College, Netanya, Israel, W. Herfort, Institut für Angewandte und Numerische Mathematik, Technische Universität Vienna, Austria; The second author would like to thank for greatful hospitality at the Bar-Ilan University, the Netanya Academic College and the Tel-Aviv University in February 2000
  • Edited by C. M. Campbell, University of St Andrews, Scotland, E. F. Robertson, University of St Andrews, Scotland, G. C. Smith, University of Bath
  • Book: Groups St Andrews 2001 in Oxford
  • Online publication: 11 January 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511542770.004
Available formats
×