Finite groups satisfying the independence property

Saul Daniel Freedman; Andrea Lucchini; Daniele Nemmi; Colva Mary Roney-Dougal

doi:10.1142/S021819672350025X

Finite groups satisfying the independence property

Saul Daniel Freedman, Andrea Lucchini^*, Daniele Nemmi, Colva Mary Roney-Dougal

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

Abstract

We say that a finite group G satisfies the independence property if, for every pair of distinct elements x and y of G, either {x, y} is contained in a minimal generating set for G or one of x and y is a power of the other. We give a complete classification of the finite groups with this property, and in particular prove that every such group is supersoluble. A key ingredient of our proof is a theorem showing that all but three finite almost simple groups H contain an element s such that the maximal subgroups of H containing s, but not containing the socle of H, are pairwise non-conjugate.

Original language	English
Pages (from-to)	509-545
Number of pages	37
Journal	International Journal of Algebra and Computation
Volume	33
Issue number	3
Early online date	29 Apr 2023
DOIs	https://doi.org/10.1142/S021819672350025X
Publication status	Published - 1 May 2023

Keywords

Generating sets
Supersoluble groups
Simple groups

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10.1142/S021819672350025X

independence and perfect rank
Copyright © 2023 the Authors. This work has been made available online in accordance with the Rights Retention Strategy This accepted manuscript is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. The final published version of this work is available at https://doi.org/10.1142/S021819672350025X.
Accepted author manuscript, 480 KBLicence: CC BY

Cite this

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title = "Finite groups satisfying the independence property",

abstract = "We say that a finite group G satisfies the independence property if, for every pair of distinct elements x and y of G, either {x, y} is contained in a minimal generating set for G or one of x and y is a power of the other. We give a complete classification of the finite groups with this property, and in particular prove that every such group is supersoluble. A key ingredient of our proof is a theorem showing that all but three finite almost simple groups H contain an element s such that the maximal subgroups of H containing s, but not containing the socle of H, are pairwise non-conjugate.",

keywords = "Generating sets, Supersoluble groups, Simple groups",

author = "Freedman, {Saul Daniel} and Andrea Lucchini and Daniele Nemmi and Roney-Dougal, {Colva Mary}",

note = "Funding: The first author was supported by a St Leonard{\textquoteright}s International Doctoral Fees Scholarship and a School of Mathematics & Statistics PhD Funding Scholarship at the University of St Andrews. The fourth author would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme “Groups, Representations and Applications: New perspectives”, where work on this paper was undertaken. This work was supported by EPSRC grant no EP/R014604/1, and also partially supported by a grant from the Simons Foundation.",

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doi = "10.1142/S021819672350025X",

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T1 - Finite groups satisfying the independence property

AU - Freedman, Saul Daniel

AU - Lucchini, Andrea

AU - Nemmi, Daniele

AU - Roney-Dougal, Colva Mary

N1 - Funding: The first author was supported by a St Leonard’s International Doctoral Fees Scholarship and a School of Mathematics & Statistics PhD Funding Scholarship at the University of St Andrews. The fourth author would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme “Groups, Representations and Applications: New perspectives”, where work on this paper was undertaken. This work was supported by EPSRC grant no EP/R014604/1, and also partially supported by a grant from the Simons Foundation.

PY - 2023/5/1

Y1 - 2023/5/1

N2 - We say that a finite group G satisfies the independence property if, for every pair of distinct elements x and y of G, either {x, y} is contained in a minimal generating set for G or one of x and y is a power of the other. We give a complete classification of the finite groups with this property, and in particular prove that every such group is supersoluble. A key ingredient of our proof is a theorem showing that all but three finite almost simple groups H contain an element s such that the maximal subgroups of H containing s, but not containing the socle of H, are pairwise non-conjugate.

AB - We say that a finite group G satisfies the independence property if, for every pair of distinct elements x and y of G, either {x, y} is contained in a minimal generating set for G or one of x and y is a power of the other. We give a complete classification of the finite groups with this property, and in particular prove that every such group is supersoluble. A key ingredient of our proof is a theorem showing that all but three finite almost simple groups H contain an element s such that the maximal subgroups of H containing s, but not containing the socle of H, are pairwise non-conjugate.

KW - Generating sets

KW - Supersoluble groups

KW - Simple groups

U2 - 10.1142/S021819672350025X

DO - 10.1142/S021819672350025X

M3 - Article

SN - 0218-1967

VL - 33

SP - 509

EP - 545

JO - International Journal of Algebra and Computation

JF - International Journal of Algebra and Computation

IS - 3

ER -

Finite groups satisfying the independence property

Abstract

Keywords

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