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When are Two Subgroups Independent?

Alexa Gopaulsingh

Abstract

Rosenmann and Ventura asked "What is the right definition of dependence of subgroups for general groups?". Here we aim to answer this question and generalise the notion first used by Rosemann. 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.

When are Two Subgroups Independent?

Abstract

Rosenmann and Ventura asked "What is the right definition of dependence of subgroups for general groups?". Here we aim to answer this question and generalise the notion first used by Rosemann. 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 and are almost disjoint if and only if , where 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.
Paper Structure (10 sections, 29 theorems)

This paper contains 10 sections, 29 theorems.

Table of Contents

  1. Introduction
  2. Subgroup Independence
  3. Definitions and Notation
  4. Main Results about Subgroup Independence
  5. Further Connections following from Subgroup Separateness
  6. Factoring the Join of Two Subgroups by One of the Subgroups
  7. Element-Writing Independence
  8. The Open Problem of Subgroup Independence
  9. What to Do if You Need to Determine if Two Subgroups Affect Each Other
  10. Acknowledgements

Key Result

Proposition 2.1

Let $A$ and $B$ be subgroups of a group. For any endomorphisms $\alpha$ and $\beta$ of $A$ and $B$ respectively, if their extension to an endomorphism exists in their join, $\gamma$ say, then for any $\prod_{i= 1}^{n}a_ib_i \in \langle A \cup B\rangle$, it is the case that $\gamma(\prod_{i=1}^{n}a_i

Theorems & Definitions (64)

  • Proposition 2.1
  • proof
  • Lemma 2.1
  • proof
  • Theorem 2.1
  • proof
  • Corollary 2.1
  • proof
  • Corollary 2.2
  • proof
  • ...and 54 more