On the minimal dimension of a finite simple group

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:171

Article

Burness, Timothy C.¹  Garonzi, Martino²  Lucchini, Andrea³ 

[1] Univ Bristol, Sch Math, Bristol BS8 1UG, Avon, England

[2] Univ Brasilia, Dept Matemat, Campus Univ Darcy Ribeiro, BR-70910900 Brasilia, DF, Brazil

[3] Univ Padua, Dipartimento Matemat Tullio Levi Civita, Via Trieste 63, I-35131 Padua, Italy

关键词: Minimal dimension;    Finite simple groups;    Maximal subgroups;    Base size;   

DOI  :  10.1016/j.jcta.2019.105175

来源: Elsevier

【 摘 要 】

Let G be a finite group and let M be a set of maximal subgroups of G. We say that M is irredundant if the intersection of the subgroups in M is not equal to the intersection of any proper subset. The minimal dimension of G, denoted Mindim(G), is the minimal size of a maximal irredundant set of maximal subgroups of G. This invariant was recently introduced by Garonzi and Lucchini and they computed the minimal dimension of the alternating groups. In this paper, we prove that Mindim(G) <= 3 for all finite simple groups, which is best possible, and we compute the exact value for all non-classical simple groups. We also introduce and study two closely related invariants denoted by alpha(G) and beta(G). Here alpha(G) (respectively beta(G)) is the minimal size of a set of maximal subgroups (respectively, conjugate maximal subgroups) of G whose intersection coincides with the Frattini subgroup of G. Evidently, Mindim(G) <= alpha(G) <= beta(G). For a simple group G we show that beta(G) <= 4 and beta(G) - alpha(G) <= 1, and both upper bounds are best possible. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free