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On the minimal dimension of a finite simple group

Research output: Contribution to journalArticle

Original languageEnglish
Article number105175
Number of pages32
JournalJournal of Combinatorial Theory, Series A
Volume171
Early online date20 Nov 2019
DOIs
DateAccepted/In press - 7 Nov 2019
DateE-pub ahead of print (current) - 20 Nov 2019
DatePublished - 1 Apr 2020

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.

    Research areas

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

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    Rights statement: This is the author accepted manuscript (AAM). The final published version (version of record) is available online via Elsevier at https://www.sciencedirect.com/science/article/pii/S0097316519301566 . Please refer to any applicable terms of use of the publisher.

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