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) <= 5 and that d(H) >= 5 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.
GENERATING MAXIMAL SUBGROUPS OF FINITE ALMOST SIMPLE GROUPS
Lucchini, ANDREA
;Marion, C;Tracey, G
2020
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) <= 5 and that d(H) >= 5 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.
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