Commensurability of automorphism groups

TL;DR

This paper develops a theory to compare the sizes of automorphism groups of modules and rings, even when infinite, motivated by number theory heuristics, with implications to be explored further.

Contribution

It introduces a new framework for understanding commensurability of automorphism groups, extending to infinite cases and motivated by number theory applications.

Findings

Established a theory of commensurability for groups, rings, and modules.

Enabled comparison of automorphism group sizes in infinite cases.

Laid groundwork for number theoretic implications in future work.

Abstract

We develop a theory of commensurability of groups, of rings, and of modules. It allows us, in certain cases, to compare sizes of automorphism groups of modules, even when those are infinite. This work is motivated by the Cohen-Lenstra heuristics on class groups. The number theoretic implications will be addressed in a separate paper.