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 | |
| 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