Geodesic
GENERATING MAXIMAL SUBGROUPS OF FINITE ALMOST SIMPLE GROUPS
Forum of Mathematics, Sigma, Tome 8 (2020)

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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. 
@article{10_1017_fms_2019_43,
     author = {ANDREA LUCCHINI and CLAUDE MARION and GARETH TRACEY},
     title = {GENERATING {MAXIMAL} {SUBGROUPS} {OF} {FINITE} {ALMOST} {SIMPLE} {GROUPS}},
     journal = {Forum of Mathematics, Sigma},
     publisher = {mathdoc},
     volume = {8},
     year = {2020},
     doi = {10.1017/fms.2019.43},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2019.43/}
}
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ANDREA LUCCHINI; CLAUDE MARION; GARETH TRACEY. GENERATING MAXIMAL SUBGROUPS OF FINITE ALMOST SIMPLE GROUPS. Forum of Mathematics, Sigma, Tome 8 (2020). doi: 10.1017/fms.2019.43

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