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GENERATING MAXIMAL SUBGROUPS OF FINITE ALMOST SIMPLE GROUPS

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

Published online by Cambridge University Press:  30 June 2020

ANDREA LUCCHINI ,
CLAUDE MARION  and
GARETH TRACEY
ANDREA LUCCHINI
Affiliation:
Dipartimento di Matematica Tullio Levi-Civita, Università degli Studi di Padova, 35121-IPadova, Italy; [email protected]
CLAUDE MARION
Affiliation:
CMUP, Departamento de Matemática, Universidade do Porto, 4169-007Porto, Portugal; [email protected]
GARETH TRACEY
Affiliation:
Department of Mathematical Sciences, University of Bath, BathBA2 7AY, UK; [email protected]

Abstract

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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.

MSC classification

Primary: 20F05: Generators, relations, and presentations
Secondary: 20D05: Finite simple groups and their classification 20E28: Maximal subgroups
Type
Algebra
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e32
Creative Commons
Creative Common License - CCCreative Common License - BY
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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