Flexibility in generating sets of finite groups

Scott Harper

doi:10.1007/s00013-021-01691-0

Flexibility in generating sets of finite groups

Scott Harper^*

^*Corresponding author for this work

School of Mathematics

Research output: Contribution to journal › Article (Academic Journal) › peer-review

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Abstract

Let G be a finite group. It has recently been proved that every nontrivial element of G is contained in a generating set of minimal size if and only if all proper quotients of G require fewer generators than G. It is natural to ask which finite groups, in addition, have the property that any two elements of G that do not generate a cyclic group can be extended to a generating set of minimal size. This note answers the question. The only such finite groups are very specific affine groups: elementary abelian groups extended by a cyclic group acting as scalars.

Original language	English
Pages (from-to)	231–237
Number of pages	7
Journal	Archiv der Mathematik
Volume	118
Issue number	3
DOIs	https://doi.org/10.1007/s00013-021-01691-0
Publication status	Published - 23 Jan 2022

Bibliographical note

Funding Information:
The author thanks the anonymous referee for their useful feedback. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/ ), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Publisher Copyright:
© 2022, The Author(s).

Keywords

Finite groups
Generating sets
Spread
Bases

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10.1007/s00013-021-01691-0

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abstract = "Let G be a finite group. It has recently been proved that every nontrivial element of G is contained in a generating set of minimal size if and only if all proper quotients of G require fewer generators than G. It is natural to ask which finite groups, in addition, have the property that any two elements of G that do not generate a cyclic group can be extended to a generating set of minimal size. This note answers the question. The only such finite groups are very specific affine groups: elementary abelian groups extended by a cyclic group acting as scalars.",

keywords = "Finite groups, Generating sets, Spread, Bases",

author = "Scott Harper",

note = "Funding Information: The author thanks the anonymous referee for their useful feedback. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/ ), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Publisher Copyright: {\textcopyright} 2022, The Author(s).",

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N2 - Let G be a finite group. It has recently been proved that every nontrivial element of G is contained in a generating set of minimal size if and only if all proper quotients of G require fewer generators than G. It is natural to ask which finite groups, in addition, have the property that any two elements of G that do not generate a cyclic group can be extended to a generating set of minimal size. This note answers the question. The only such finite groups are very specific affine groups: elementary abelian groups extended by a cyclic group acting as scalars.

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KW - Finite groups

KW - Generating sets

KW - Spread

KW - Bases

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DO - 10.1007/s00013-021-01691-0

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ER -

Flexibility in generating sets of finite groups

Abstract

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