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

TL;DR

This paper characterizes centralizer-like subgroups related to the $n$-Engel words in direct product groups, showing the structure of right $n$-Engel elements in such products and proposing future research directions.

Contribution

It establishes the equivalence of right $n$-Engel elements in a direct product with the product of the individual sets, extending prior work on centralizer-like subgroups.

Findings

Right $n$-Engel elements in direct products decompose into products of individual sets.

The set of right $n$-Engel elements in a direct product equals the product of the sets in each factor.

New questions and future research directions are proposed.

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.