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

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.