When are Two Subgroups Independent?

TL;DR

This paper explores a category-theoretic definition of subgroup independence in groups, providing necessary and sufficient conditions, and introduces an algorithm to determine independence in many cases.

Contribution

It introduces a new, general notion of subgroup independence based on endomorphism extension, expanding the understanding beyond almost disjointness.

Findings

Almost disjointness does not imply independence.

Necessary and sufficient conditions for subgroup independence are identified.

A heuristic algorithm for deciding subgroup independence is proposed.

Abstract

Rosenmann and Ventura asked "What is the right definition of dependence of subgroups for general groups?". Here we aim to answer this question. We consider a definition of subgroup independence which is a special case of a category-theoretic one. It is that: Two subgroups of a group are independent if and only if any two endomorphisms, one acting on each subgroup, can be extended to an endomorphism of the group generated by these subgroups. This definition helps to illuminate that the usual condition of almost disjointness of subgroups (two subgroups $A$ and $B$ are almost disjoint if and only if $A \cap B = \{e\}$, where $e$ is the identity element) is not enough to force independence and here we find necessary and (different) sufficient conditions for subgroup independence. The aim of this note is to introduce this general notion of subgroup independence to the group theory community and to pose the open question of its characterisation. We present the partial results known up to this point. Moreover, we use the progress made so far to give a heuristic algorithm that decides subgroup independence for many cases.