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arXiv · 2602.15463

Subgroups with all finite lifts isomorphic are conjugate

Ido Karshon·Alexander Lubotzky·D. B. McReynolds·Alan W. Reid·Mark Shusterman
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Abstract

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.

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BibTeXRIS

Ido Karshon, Alexander Lubotzky, D. B. McReynolds, Alan W. Reid, Mark Shusterman. 2026-02-17. Subgroups with all finite lifts isomorphic are conjugate. https://arxiv.org/abs/2602.15463

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