Subgroups with all finite lifts isomorphic are conjugate

TL;DR

The paper demonstrates that non-conjugate subgroups with all finite lifts isomorphic can be distinguished via finite extensions, impacting the understanding of subgroup isomorphism and related geometric and algebraic structures.

Contribution

It shows the existence of finite extensions where pre-images of non-conjugate subgroups are not isomorphic, answering a specific question and connecting to broader mathematical areas.

Findings

Existence of finite extensions distinguishing non-conjugate subgroups

Counterexamples to $\

Z$-coset equivalence implying isomorphism

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.