The Lattice of Group Topologies
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866911148297682944 |
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| author | Peng, Dekui |
| author_facet | Peng, Dekui |
| contents | For an infinite group $G$, the poset $\mathcal{L}_G$ of group topologies constitutes a complete lattice. Although $\mathcal{L}_G$ is modular when $G$ is abelian, this property fails to persist for nilpotent groups. Extending Arnautov's 2010 work on the semi-modularity of $\mathcal{L}_G$ for nilpotent groups, we present an alternative proof with enhanced structural clarity. Additionally, we resolve two open questions from the Kourovka Notebook regarding lattice-theoretic properties of $\mathcal{L}_G$: (1) explicit construction of a countably infinite non-abelian nilpotent group with modular topology lattice, and (2) establishing the absence of property $P_2$ in infinite abelian groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_08269 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Lattice of Group Topologies Peng, Dekui General Topology Group Theory 54A10, 20F18 For an infinite group $G$, the poset $\mathcal{L}_G$ of group topologies constitutes a complete lattice. Although $\mathcal{L}_G$ is modular when $G$ is abelian, this property fails to persist for nilpotent groups. Extending Arnautov's 2010 work on the semi-modularity of $\mathcal{L}_G$ for nilpotent groups, we present an alternative proof with enhanced structural clarity. Additionally, we resolve two open questions from the Kourovka Notebook regarding lattice-theoretic properties of $\mathcal{L}_G$: (1) explicit construction of a countably infinite non-abelian nilpotent group with modular topology lattice, and (2) establishing the absence of property $P_2$ in infinite abelian groups. |
| title | The Lattice of Group Topologies |
| topic | General Topology Group Theory 54A10, 20F18 |
| url | https://arxiv.org/abs/2310.08269 |