Left 3-Engel Elements

Student thesis: Doctoral ThesisPhD

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 Award2 Nov 2022
Original languageEnglish
Awarding Institution
  • University of Bath
SponsorsEPSRC & Mandy Norton Scholarship
SupervisorGunnar Traustason (Supervisor) & Alastair King (Advisor)

Keywords

  • Engel groups
  • Engel elements
  • Left 3-Engel
  • Lie ring
  • Lie algebra
  • Hirsch-Plotkin radical

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