PreprintCC BY 4.0arXiv
Maximal subgroups of homeomorphism groups
Abstract
We show that the homeomorphism groups of the following spaces have precisely 22ℵ0 maximal subgroups: the rational numbers ℚ, the Baire space ℕℕ, the space ℕ× 2ℕ where 2ℕ is the Cantor set, the ordinal ω2 under its order topology, and the Sorgenfrey line 𝕊. More generally, we find sufficient conditions on a group G acting on a topological space which imply that G has at least 22ℵ0 maximal subgroups. Moreover, if the groups Homeo(ℚ) and Homeo(ℕN) are equipped with the pointwise topology, then it is shown that Homeo(ℕN) has precisely 2ℵ0 open maximal subgroups, and Homeo(ℚ) has precisely ℵ0 open maximal subgroups and 2ℵ0 closed maximal subgroups.
Related
- A note on maximal subgroups of countable groups
- On second maximal subgroups of finite groups
- Maximal ranks of free abelian subgroups of generalized cactus groups associated with finite Coxeter systems
- Orbits of maximal and submaximal dimension for Sylow p-subgroups of finite classical groups
- Finite Groups with Few Almost Maximal Subgroups
BibTeX
@misc{bardyla2026maximal,
title = {{Maximal subgroups of homeomorphism groups}},
author = {S. Bardyla and L. Elliott and Y. Péresse},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.28211},
eprint = {2608.28211},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28211},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Maximal subgroups of homeomorphism groups
AU - S. Bardyla
AU - L. Elliott
AU - Y. Péresse
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.28211
UR - https://arxiv.org/abs/2608.28211
AB - We show that the homeomorphism groups of the following spaces have precisely 2^2^ℵ_0 maximal subgroups: the rational numbers ℚ, the Baire space ℕ^ℕ, the space ℕ× 2^ℕ where 2^ℕ is the Cantor set, the ordinal ω^2 under its order topology, and the Sorgenfrey line 𝕊. More generally, we find sufficient conditions on a group G acting on a topological space which imply that G has at least 2^2^ℵ_0 maximal subgroups. Moreover, if the groups Homeo(ℚ) and Homeo(ℕ^N) are equipped with the pointwise topology, then it is shown that Homeo(ℕ^N) has precisely 2^ℵ_0 open maximal subgroups, and Homeo(ℚ) has precisely ℵ_0 open maximal subgroups and 2^ℵ_0 closed maximal subgroups.
ER -