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

Pure Mathematics

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

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
Early online date	23 Jan 2022
DOIs	https://doi.org/10.1007/s00013-021-01691-0
Publication status	Published - Mar 2022

Keywords

Finite grouops
Generating sets
Spread
Bases

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Cite this

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KW - Spread

KW - Bases

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Flexibility in generating sets of finite groups

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