A polynomial bound for the number of maximal systems of imprimitivity of a finite transitive permutation group

  • 1 Dipartimento di Matematica Pura e Applicata, University of Milano-Bicocca, Via Cozzi 55, 20126, Milano, Italy
  • 2 Dipartimento di Matematica “Tullio Levi-Civita”, University of Padova, Via Trieste 53, 35121, Padova, Italy
Andrea LucchiniORCID iD: https://orcid.org/0000-0002-2134-4991, Mariapia MoscatielloORCID iD: https://orcid.org/0000-0002-6493-6858 and Pablo SpigaORCID iD: https://orcid.org/0000-0002-0157-7405

Abstract

We show that there exists a constant a such that, for every subgroup H of a finite group G, the number of maximal subgroups of G containing H is bounded above by a|G:H|3/2. In particular, a transitive permutation group of degree n has at most an3/2 maximal systems of imprimitivity. When G is soluble, generalizing a classic result of Tim Wall, we prove a much stronger bound, that is, the number of maximal subgroups of G containing H is at most |G:H|-1.

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