Abstract
Houghton's groups $H_2, H_3, \ldots$ have been studied in many contexts, and various results exist for their finite index subgroups. In this note we describe all of the finite index subgroups of each Houghton group, and their isomorphism types. Using the standard notation that $d(G)$ denotes the minimal size of generating set for $G$ we then show, for each $n\in \{2, 3,\ldots\}$ and $U$ of finite index in $H_n$, that $d(U)\in\{d(H_n), d(H_n)+1\}$ and characterise when each of these cases occurs.
| Original language | English |
|---|---|
| Pages (from-to) | 113-121 |
| Number of pages | 9 |
| Journal | Archiv der Mathematik |
| Volume | 118 |
| Issue number | 2 |
| DOIs | |
| Publication status | Published - 29 Jan 2022 |
Bibliographical note
Funding Information:I thank the anonymous referee and the editor for their excellent comments. 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
- Generation
- Generation of finite index subgroups
- Structure of finite index subgroups
- Infinite groups
- Houghton groups
- Permutation groups
- Highly transitive groups