Subgroups with all finite lifts isomorphic are conjugate

TL;DR

This work shows that non-conjugate subgroups of a finite group $G$ can have non-isomorphic pre-images in a finite extension, by constructing a finite $\widetilde{G}$ with a surjection $\varphi: \widetilde{G}\to G$ where $\varphi^{-1}(G_1)$ and $\varphi^{-1}(G_2)$ are not isomorphic for all non-conjugate $G_1,G_2$. The construction leverages a realization $G \cong \operatorname{Out}(N)$ from a finite $N$ (via Sambale/Cornulier) and yields a kernel that can be chosen supersolvable, preserving solvability when $G$ is solvable. A key application is that $\mathbb{Z}$-coset equivalent subgroups need not be isomorphic, answering a question of Prasad in the negative, and the paper situates these results within profinite rigidity, anabelian geometry, and related areas. The authors also provide a concrete computational example using a $\mathrm{PSL}(2,29)$-based construction to illustrate the finite-quotient phenomenon and discuss broader implications and potential extensions.

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.

Theorems & Definitions (19)

• Theorem 1.1
• Theorem 1.2: Y. Cornulier
• Theorem 1.4
• Corollary 1.6
• Theorem 1.7
• Lemma 2.1
• proof
• Proposition 3.1
• proof
• proof : Proof of Corollary \ref{['CorProfComp']}