c-Sections, solvability and large subgroups of finite groups

Authors

  • Barbara Baumeister Universitaet Bielefeld
  • Gil Kaplan The Academic College of Tel-Aviv-Yaffo

Keywords:

$c$-section, solvability, maximal subgroup, large subgroup

Abstract

c-Sections of maximal subgroups in a finite group and their relation to solvability were extensively researched in recent years (see [SW], [W] and [LS]). A fundamental result [W] is that a finite group is solvable \iff\ the $c$-sections of all its maximal subgroups are trivial. In this paper we prove (Theorem 1.2), 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 opposite direction 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 [L]. DOI: 10.1017/S0004972712000081

Author Biographies

Barbara Baumeister, Universitaet Bielefeld

Fakultaet fuer Mathematik

Gil Kaplan, The Academic College of Tel-Aviv-Yaffo

School of Computer Sciences

Published

2012-08-27

Issue

Section

Articles