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Boolean lattices in finite alternating and symmetric groups

Part of: Permutation groups

Published online by Cambridge University Press:  13 November 2020

Andrea Lucchini Open the ORCID record for Andrea Lucchini [Opens in a new window] ,
Mariapia Moscatiello ,
Sebastien Palcoux Open the ORCID record for Sebastien Palcoux [Opens in a new window]  and
Pablo Spiga Open the ORCID record for Pablo Spiga [Opens in a new window]
Andrea Lucchini
Affiliation:
Dipartimento di Matematica “Tullio Levi-Civita”, University of Padova, Via Trieste 53, 35121 Padova, Italy; E-mail: [email protected]
Mariapia Moscatiello
Affiliation:
Dipartimento di Matematica “Tullio Levi-Civita”, University of Padova, Via Trieste 53, 35121 Padova, Italy; E-mail: [email protected]
Sebastien Palcoux
Affiliation:
Yau Mathematical Sciences Center, Tsinghua University, Beijing, China; E-mail: [email protected]
Pablo Spiga
Affiliation:
Dipartimento di Matematica e Applicazioni, University of Milano-Bicocca, Via Cozzi 55, 20125 Milano, Italy; E-mail: [email protected]

Abstract

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Given a group G and a subgroup H, we let $\mathcal {O}_G(H)$ denote the lattice of subgroups of G containing H. This article provides a classification of the subgroups H of G such that $\mathcal {O}_{G}(H)$ is Boolean of rank at least $3$ when G is a finite alternating or symmetric group. Besides some sporadic examples and some twisted versions, there are two different types of such lattices. One type arises by taking stabilisers of chains of regular partitions, and the other arises by taking stabilisers of chains of regular product structures. As an application, we prove in this case a conjecture on Boolean overgroup lattices related to the dual Ore’s theorem and to a problem of Kenneth Brown.

MSC classification

Primary: 20B25: Finite automorphism groups of algebraic, geometric, or combinatorial structures
Type
Algebra
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e55
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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