We construct a Lie relator which is not an identical Lie relator. 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<sup>j</sup>, L<sup>i</sup>, L<sup>k</sup><./em>] is torsion free. Also, we prove that if 1 < i ≤ j ≤ 2i and j ≤ k ≤ l ≤ i + j then L/[L<sup>j</sup>, L<sup>i</sup>, L<sup>k</sup>, L<sup>l</sup>] is torsion free. We then prove that the analogous groups, namely F/[γ<sub>j</sub>(F),γ<sub>i</sub>(F),γ<sub>k</sub>(F)] and F/[γ<sub>j</sub>(F),γ<sub>i</sub>(F),γ<sub>k</sub>(F),γ<sub>l</sub>(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<sup>j</sup>, L<sup>i</sup>, L<sup>k</sup>] 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<sup>T(m,k)</sup>. We prove that if G is a finite m-generator group of exponent 8 then |G| ≤ T(m, 7<sup>471</sup>), improving upon the best previously known bound of T(m, 8<sup>88</sup>).
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:343302 |
| Date | January 2000 |
| Creators | Groves, Daniel |
| Contributors | Vaughan-Lee, Michael |
| Publisher | University of Oxford |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://ora.ox.ac.uk/objects/uuid:4b5479ad-30ac-4ad6-98a3-51484095868b |
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