Problems in Lie rings and groups

Thesis

We construct a Lie relator which is not an identical Lie relator in the variety 𝔓₄𝔓₂. This is the first known example of a non-identical Lie relator.

Next we consider the existence of torsion in outer commutator groups. Let L be a free Lie ring. Suppose that 1 < i ≤ j ≤ 2i and i ≤ k ≤ i + j + 1. We prove that L/[L^j, Lⁱ, L^k] is torsion free. Also, we prove that if 1 < i ≤ j ≤ 2i and j ≤ k ≤ l ≤ i + j then L/[L^j, Lⁱ, L^k, L^l

] is torsion free. We then prove that the analogous groups, namely F/[γ_j(F),γ_i(F),γ_k(F)] and F/[γ_j(F),γ_i(F),γ_k(F),γ_l(F)] (under the same conditions for i, j, k and i, j, k, l respectively), are residually nilpotent and torsion free. We prove the existence of 2-torsion in the Lie rings L/[L^j, Lⁱ, L^k] when 1 ≤ k < i,j ≤ 5, and thus show that our methods do not work in these cases.

Finally, we consider the order of finite groups of exponent 8. For m ≥ 2, we define the function T(m,n)

by T(m,1) = m and T(m,k + 1) = m^T(m,k). We prove that if G is a finite m-generator group of exponent 8 then |G| ≤ T(m, 7⁴⁷¹), improving upon the best previously known bound of T(m, 8⁸⁸).

Authors: Groves, D; Daniel Groves

• Publication date: 2000
• DOI: 10.5287/ora-goeoxp4qd
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Subjects: Lie algebras; Lie groups