Flexibility in generating sets of finite groups
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.
Citation
Harper , S 2022 , ' Flexibility in generating sets of finite groups ' , Archiv der Mathematik , vol. 118 , pp. 231-237 . https://doi.org/10.1007/s00013-021-01691-0
Publication
Archiv der Mathematik
Status
Peer reviewed
ISSN
0003-889XType
Journal article
Collections
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