Left 3-Engel Elements

Abstract

In this thesis we focus on results concerning left 3-Engel elements. We will first record some basic definitions and elementary results about groups and Lie algebras. We then focus on Engel groups and Engel elements. In particular, we generalise the construction given in the paper "A left 3-Engel element whose normal closure is not nilpotent" by Noce, Tracey and Traustason, by giving an infinite family of examples of a locally finite 2-group G containing a left 3-Engel element x where the normal closure of x in G, is not nilpotent. The construction is based on a family of Lie algebras that are of interest in their own right given in the paper "Engel Lie-algebras" by Traustason, and make use of a classical theorem of Lucas, regarding when the binomial coefficient m choose n is even. We then proceed to the main result of this thesis. We show that in the case of any odd prime p, we can give an example of a locally finite p-group G containing a left 3-Engel element x where the normal closure of x in G is not nilpotent.

Date of Award	2 Nov 2022
Original language	English
Awarding Institution	University of Bath
Sponsors	EPSRC - EU & Mandy Norton Scholarship
Supervisor	Gunnar Traustason (Supervisor) & Alastair King (Advisor)