Subgroups with all finite lifts isomorphic are conjugate
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866918483879526400 |
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| author | Karshon, Ido Lubotzky, Alexander McReynolds, D. B. Reid, Alan W. Shusterman, Mark |
| author_facet | Karshon, Ido Lubotzky, Alexander McReynolds, D. B. Reid, Alan W. Shusterman, Mark |
| contents | We show that for non-conjugate subgroups $G_1$ and $G_2$ of a finite group $G$ there exists an extension of $G$ (by a finite group) in which the pre-images of $G_1$ and $G_2$ are not isomorphic. This allows us to show that $\mathbb Z$-coset equivalent subgroups of a finite group are not necessarily isomorphic, answering a question of Dipendra Prasad. We also indicate connections to profinite rigidity, anabelian geometry, mapping class groups, and non-arithmetic lattices in Lie groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_15463 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Subgroups with all finite lifts isomorphic are conjugate Karshon, Ido Lubotzky, Alexander McReynolds, D. B. Reid, Alan W. Shusterman, Mark Group Theory Algebraic Geometry Geometric Topology Number Theory We show that for non-conjugate subgroups $G_1$ and $G_2$ of a finite group $G$ there exists an extension of $G$ (by a finite group) in which the pre-images of $G_1$ and $G_2$ are not isomorphic. This allows us to show that $\mathbb Z$-coset equivalent subgroups of a finite group are not necessarily isomorphic, answering a question of Dipendra Prasad. We also indicate connections to profinite rigidity, anabelian geometry, mapping class groups, and non-arithmetic lattices in Lie groups. |
| title | Subgroups with all finite lifts isomorphic are conjugate |
| topic | Group Theory Algebraic Geometry Geometric Topology Number Theory |
| url | https://arxiv.org/abs/2602.15463 |