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Centralizer-like Subgroups Associated with the $n$-Engel Words Inside of Direct Product Groups

Bridget Lee, Maggie Reardon, Faculty Mentor Dandrielle Lewis

Abstract

This research provides a characterization of centralizer-like subgroups associated with the $n$-Engel word in a direct product of groups. Specifically, properties of the set of right $n$-Engel elements inside of direct products are explored. A proof is given to demonstrate the equivalence between the set of right $n$-Engel elements of a direct product of two groups and a direct product of the set of right $n$-Engel elements of each direct factor. This work was inspired by the study of centralizer-like subgroups in paper written by Luise-Charlotte Kappe and Patrick Ratchford. We present additional questions explored during this project, and we propose future research possibilities.

Centralizer-like Subgroups Associated with the $n$-Engel Words Inside of Direct Product Groups

Abstract

This research provides a characterization of centralizer-like subgroups associated with the -Engel word in a direct product of groups. Specifically, properties of the set of right -Engel elements inside of direct products are explored. A proof is given to demonstrate the equivalence between the set of right -Engel elements of a direct product of two groups and a direct product of the set of right -Engel elements of each direct factor. This work was inspired by the study of centralizer-like subgroups in paper written by Luise-Charlotte Kappe and Patrick Ratchford. We present additional questions explored during this project, and we propose future research possibilities.

Paper Structure

This paper contains 10 sections, 5 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Terminology and Background
  3. General Results
  4. The Main Theorems
  5. Continuing Research
  6. Attempted Proofs of Conjecture 6
  7. Property $1$
  8. Property $2$
  9. Conjecture 6 Example
  10. Conclusion

Key Result

Proposition 1

Kappe Let $G$ be a group, $g \in G$ and $g^G$ the normal closure of $g$ in $G$. Then,

Theorems & Definitions (19)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Definition 9
  • Definition 10
  • ...and 9 more