Some examples of invariably generated groups
Some examples of invariably generated groups
A group G is invariably generated (IG) if there is a subset S ⊆ G such that for every subset S′ ⊆ G, obtained from S by replacing each element with a conjugate, S′ generates G. Likewise, G is finitely invariably generated (FIG) if, in addition, one can choose such a subset S to be finite. In this note we construct a FIG group G with an index 2 subgroup N < G such that N is not IG. This shows that neither property IG nor FIG is stable under passing to subgroups of finite index, answering questions of Wiegold and Kantor-Lubotzky-Shalev. We also produce first examples of finitely generated IG groups that are not FIG, answering a question of Cox.
Invariably generated, finite index subgroups, finitely invariably generated
231-257
Minasyan, Ashot
3de640f5-d07b-461f-b130-5b1270bfdb3d
6 October 2021
Minasyan, Ashot
3de640f5-d07b-461f-b130-5b1270bfdb3d
Abstract
A group G is invariably generated (IG) if there is a subset S ⊆ G such that for every subset S′ ⊆ G, obtained from S by replacing each element with a conjugate, S′ generates G. Likewise, G is finitely invariably generated (FIG) if, in addition, one can choose such a subset S to be finite. In this note we construct a FIG group G with an index 2 subgroup N < G such that N is not IG. This shows that neither property IG nor FIG is stable under passing to subgroups of finite index, answering questions of Wiegold and Kantor-Lubotzky-Shalev. We also produce first examples of finitely generated IG groups that are not FIG, answering a question of Cox.
Text
invar_gen-05
- Accepted Manuscript
More information
Accepted/In Press date: 14 September 2020
e-pub ahead of print date: 6 October 2021
Published date: 6 October 2021
Keywords:
Invariably generated, finite index subgroups, finitely invariably generated
Identifiers
Local EPrints ID: 443938
URI: http://eprints.soton.ac.uk/id/eprint/443938
ISSN: 0021-2172
PURE UUID: 5000511a-6fe3-4f89-9048-34c130b31708
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Date deposited: 17 Sep 2020 16:31
Last modified: 17 Mar 2024 05:55
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