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GENERATING MAXIMAL SUBGROUPS OF FINITE ALMOST SIMPLE GROUPS

Published online by Cambridge University Press:  30 June 2020

ANDREA LUCCHINI
Affiliation:
Dipartimento di Matematica Tullio Levi-Civita, Università degli Studi di Padova, 35121-IPadova, Italy; lucchini@math.unipd.it
CLAUDE MARION
Affiliation:
CMUP, Departamento de Matemática, Universidade do Porto, 4169-007Porto, Portugal; claude.marion@fc.up.pt
GARETH TRACEY
Affiliation:
Department of Mathematical Sciences, University of Bath, BathBA2 7AY, UK; gmt29@bath.ac.uk

Abstract

For a finite group $G$ , let $d(G)$ denote the minimal number of elements required to generate $G$ . In this paper, we prove sharp upper bounds on $d(H)$ whenever $H$ is a maximal subgroup of a finite almost simple group. In particular, we show that $d(H)\leqslant 5$ and that $d(H)\geqslant 4$ if and only if $H$ occurs in a known list. This improves a result of Burness, Liebeck and Shalev. The method involves the theory of crowns in finite groups.

Type
Algebra
Creative Commons
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2020

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