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c-SECTIONS, SOLVABILITY AND LARGE SUBGROUPS OF FINITE GROUPS

Part of: Structure and classification of infinite or finite groups Representation theory of 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: baumeist@math.uni-bielefeld.de)
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: baumeist@math.uni-bielefeld.de
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Abstract

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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

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