Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-29T11:02:50.026Z Has data issue: false hasContentIssue false
  • English
  • Français

c-SECTIONS, SOLVABILITY AND LARGE SUBGROUPS OF FINITE GROUPS

Part of: Representation theory of groups Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  03 April 2012

BARBARA BAUMEISTER  and
GIL KAPLAN
BARBARA BAUMEISTER*
Affiliation:
Fakultät für Mathematik, Universität Bielefeld, Postfach 10 01 31, D-33501 Bielefeld, Germany (email: [email protected])
GIL KAPLAN
Affiliation:
School of Computer Sciences, The Academic College of Tel-Aviv-Yaffo, 2 Rabenu Yeruham st., Tel-Aviv 64044, Israel
*
✉For correspondence; e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

c-Sections of maximal subgroups in a finite group and their relation to solvability have been extensively researched in recent years. A fundamental result due to Wang [‘C-normality of groups and its properties’, J. Algebra 180 (1998), 954–965] is that a finite group is solvable if and only if the c-sections of all its maximal subgroups are trivial. In this paper we prove that if for each maximal subgroup of a finite group G, the corresponding c-section order is smaller than the index of the maximal subgroup, then each composition factor of G is either cyclic or isomorphic to the O’Nan sporadic group (the converse does not hold). Furthermore, by a certain ‘refining’ of the latter theorem we obtain an equivalent condition for solvability. Finally, we provide an existence result for large subgroups in the sense of Lev [‘On large subgroups of finite groups’ J. Algebra 152 (1992), 434–438].

Keywords

c-sectionsolvabilitymaximal subgrouplarge subgroup

MSC classification

Secondary: 20E34: General structure theorems 20D05: Finite simple groups and their classification 20D10: Solvable groups, theory of formations, Schunck classes, Fitting classes, $pi$-length, ranks
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 86 , Issue 2 , October 2012 , pp. 291 - 302
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

References

[1]Carter, R., Simple Groups of Lie Type (Wiley, London, 1972).Google Scholar
[2]Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A. and Wilson, R. A., Atlas of Finite Groups (Clarendon, Oxford, 1985).Google Scholar
[3]Lev, A., ‘On large subgroups of finite groups’, J. Algebra 152 (1992), 434–438.CrossRefGoogle Scholar
[4]Li, S. and Shi, W., ‘A note on the solvability of groups’, arXiv:math/0509377v2 (2005).Google Scholar
[5]Shirong, L. and Wang, Y., ‘On c-section and c-index of finite groups’, J. Pure Appl. Algebra 151 (2000), 300–319.CrossRefGoogle Scholar
[6]Wang, Y., ‘C-normality of groups and its properties’, J. Algebra 180 (1998), 954–965.CrossRefGoogle Scholar
[7]Wilson, R. A., ‘The maximal subgroups of the O’Nan group’, J. Algebra 97 (1985), 467–473.CrossRefGoogle Scholar