The intersection graph of a finite simple group has diameter at most 5

Saul D. Freedman

doi:10.1007/s00013-021-01583-3

The intersection graph of a finite simple group has diameter at most 5

Saul D. Freedman

Pure Mathematics

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Abstract

Let G be a non-abelian finite simple group. In addition, let Δ_G be the intersection graph of G, whose vertices are the proper non-trivial subgroups of G, with distinct subgroups joined by an edge if and only if they intersect non-trivially. We prove that the diameter of Δ_G has a tight upper bound of 5, thereby resolving a question posed by Shen (Czechoslov Math J 60(4):945–950, 2010). Furthermore, a diameter of 5 is achieved only by the baby monster group and certain unitary groups of odd prime dimension.

Original language	English
Pages (from-to)	1-7
Number of pages	7
Journal	Archiv der Mathematik
Volume	117
Issue number	1
Early online date	13 Feb 2021
DOIs	https://doi.org/10.1007/s00013-021-01583-3
Publication status	Published - Jul 2021

Keywords

Intersection graph
Simple group
Subgroups

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10.1007/s00013-021-01583-3Licence: CC BY

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Copyright © 2021 The Author(s). Open Access. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
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Cite this

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abstract = "Let G be a non-abelian finite simple group. In addition, let ΔG be the intersection graph of G, whose vertices are the proper non-trivial subgroups of G, with distinct subgroups joined by an edge if and only if they intersect non-trivially. We prove that the diameter of ΔG has a tight upper bound of 5, thereby resolving a question posed by Shen (Czechoslov Math J 60(4):945–950, 2010). Furthermore, a diameter of 5 is achieved only by the baby monster group and certain unitary groups of odd prime dimension.",

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N2 - Let G be a non-abelian finite simple group. In addition, let ΔG be the intersection graph of G, whose vertices are the proper non-trivial subgroups of G, with distinct subgroups joined by an edge if and only if they intersect non-trivially. We prove that the diameter of ΔG has a tight upper bound of 5, thereby resolving a question posed by Shen (Czechoslov Math J 60(4):945–950, 2010). Furthermore, a diameter of 5 is achieved only by the baby monster group and certain unitary groups of odd prime dimension.

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KW - Simple group

KW - Subgroups

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The intersection graph of a finite simple group has diameter at most 5

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