Subgroups with all finite lifts isomorphic are conjugate

2026/02/28 by Ido Karshon, Alexander Lubotzky, D. B. McReynolds +2
Mathematics · #math.GR #math.AG #math.GT #math.NT

paper · pdf

v3. Minor changes. 10 pages

arxiv created 2026/08/01 · arxiv updated 2026/08/04

Abstract

We show that for non-conjugate subgroups G₁ and G₂ of a finite group G there exists an extension of G (by a finite group) in which the pre-images of G₁ and G₂ 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.

Citations

• Characterizing inner automorphisms and realizing outer automorphisms
• Integral Gassman equivalence of algebraic and hyperbolic manifolds
• A refined notion of arithmetically equivalent number fields, and curves with isomorphic Jacobians
• Non-arithmetic groups in Lobachevsky spaces