On the minimal dimension of a finite simple group

Timothy C. Burness, Martino Garonzi, Andrea Lucchini

Research output: Contribution to journalArticle (Academic Journal)peer-review

Abstract

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 α(G) and β(G). Here α(G) (respectively β(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)⩽α(G)⩽β(G). For a simple group G we show that β(G)⩽4 and β(G)−α(G)⩽1, and both upper bounds are best possible.

Original languageEnglish
Article number105175
Number of pages32
JournalJournal of Combinatorial Theory, Series A
Volume171
Early online date20 Nov 2019
DOIs
Publication statusPublished - 1 Apr 2020

Keywords

  • minimal dimension
  • finite simple groups
  • maximal subgroups
  • base size

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