Problems in Lie rings and groups
<p>We construct a Lie relator which is not an identical Lie relator in the variety 𝔓<sub>4</sub>𝔓<sub>2</sub>. This is the first known example of a non-identical Lie relator.</p> <p>Next we consider the existence of torsion in outer commutator groups. Let &l...
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| Format: | Thesis |
| Language: | English |
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2000
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| _version_ | 1797067082210213888 |
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| author | Groves, D Daniel Groves |
| author2 | Vaughan-Lee, M |
| author_facet | Vaughan-Lee, M Groves, D Daniel Groves |
| author_sort | Groves, D |
| collection | OXFORD |
| description | <p>We construct a Lie relator which is not an identical Lie relator in the variety 𝔓<sub>4</sub>𝔓<sub>2</sub>. This is the first known example of a non-identical Lie relator.</p> <p>Next we consider the existence of torsion in outer commutator groups. Let <em>L</em> be a free Lie ring. Suppose that 1 < <em>i</em> ≤ <em>j</em> ≤ <em>2i</em> and <em>i</em> ≤ <em>k</em> ≤ <em>i</em> + <em>j</em> + 1. We prove that <em>L</em>/[<em>L<sup>j</sup></em>, <em>L<sup>i</sup></em>, <em>L<sup>k</sup></em>] is torsion free. Also, we prove that if 1 < <em>i</em> ≤ <em>j</em> ≤ 2<em>i</em> and <em>j</em> ≤ <em>k</em> ≤ <em>l</em> ≤ <em>i</em> + <em>j</em> then <em>L</em>/[<em>L<sup>j</sup></em>, <em>L<sup>i</sup></em>, <em>L<sup>k</sup></em>, <em>L<sup>l</sup></em></p>] is torsion free. We then prove that the analogous groups, namely <em>F</em>/[γ<sub><em>j</em></sub>(<em>F</em>),γ<sub><em>i</em></sub>(<em>F</em>),γ<sub><em>k</em></sub>(<em>F</em>)] and <em>F</em>/[γ<sub><em>j</em></sub>(<em>F</em>),γ<sub><em>i</em></sub>(<em>F</em>),γ<sub><em>k</em></sub>(<em>F</em>),γ<sub><em>l</em></sub>(<em>F</em>)] (under the same conditions for <em>i, j, k</em> and <em>i, j, k, l</em> respectively), are residually nilpotent and torsion free. We prove the existence of 2-torsion in the Lie rings <em>L</em>/[<em>L<sup>j</sup>, L<sup>i</sup>, L<sup>k</sup></em>] when 1 ≤ <em>k</em> < <em>i</em>,<em>j</em> ≤ 5, and thus show that our methods do not work in these cases. <p>Finally, we consider the order of finite groups of exponent 8. For <em>m</em> ≥ 2, we define the function <em>T</em>(<em>m</em>,<em>n</em>)</p> by <em>T(m</em>,1) = <em>m</em> and <em>T</em>(<em>m</em>,<em>k</em> + 1) = <em>m</em><sup><em>T</em>(<em>m</em>,<em>k</em>)</sup>. We prove that if <em>G</em> is a finite <em>m</em>-generator group of exponent 8 then |<em>G</em>| ≤ <em>T(m</em>, 7<sup>471</sup>), improving upon the best previously known bound of <em>T(m</em>, 8<sup>8<sup>8</sup></sup>). |
| first_indexed | 2024-03-06T21:51:13Z |
| format | Thesis |
| id | oxford-uuid:4b5479ad-30ac-4ad6-98a3-51484095868b |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T21:51:13Z |
| publishDate | 2000 |
| record_format | dspace |
| spelling | oxford-uuid:4b5479ad-30ac-4ad6-98a3-51484095868b2022-03-26T15:42:57ZProblems in Lie rings and groupsThesishttp://purl.org/coar/resource_type/c_db06uuid:4b5479ad-30ac-4ad6-98a3-51484095868bLie algebrasLie groupsEnglishPolonsky Theses Digitisation Project2000Groves, DDaniel GrovesVaughan-Lee, M<p>We construct a Lie relator which is not an identical Lie relator in the variety 𝔓<sub>4</sub>𝔓<sub>2</sub>. This is the first known example of a non-identical Lie relator.</p> <p>Next we consider the existence of torsion in outer commutator groups. Let <em>L</em> be a free Lie ring. Suppose that 1 < <em>i</em> ≤ <em>j</em> ≤ <em>2i</em> and <em>i</em> ≤ <em>k</em> ≤ <em>i</em> + <em>j</em> + 1. We prove that <em>L</em>/[<em>L<sup>j</sup></em>, <em>L<sup>i</sup></em>, <em>L<sup>k</sup></em>] is torsion free. Also, we prove that if 1 < <em>i</em> ≤ <em>j</em> ≤ 2<em>i</em> and <em>j</em> ≤ <em>k</em> ≤ <em>l</em> ≤ <em>i</em> + <em>j</em> then <em>L</em>/[<em>L<sup>j</sup></em>, <em>L<sup>i</sup></em>, <em>L<sup>k</sup></em>, <em>L<sup>l</sup></em></p>] is torsion free. We then prove that the analogous groups, namely <em>F</em>/[γ<sub><em>j</em></sub>(<em>F</em>),γ<sub><em>i</em></sub>(<em>F</em>),γ<sub><em>k</em></sub>(<em>F</em>)] and <em>F</em>/[γ<sub><em>j</em></sub>(<em>F</em>),γ<sub><em>i</em></sub>(<em>F</em>),γ<sub><em>k</em></sub>(<em>F</em>),γ<sub><em>l</em></sub>(<em>F</em>)] (under the same conditions for <em>i, j, k</em> and <em>i, j, k, l</em> respectively), are residually nilpotent and torsion free. We prove the existence of 2-torsion in the Lie rings <em>L</em>/[<em>L<sup>j</sup>, L<sup>i</sup>, L<sup>k</sup></em>] when 1 ≤ <em>k</em> < <em>i</em>,<em>j</em> ≤ 5, and thus show that our methods do not work in these cases. <p>Finally, we consider the order of finite groups of exponent 8. For <em>m</em> ≥ 2, we define the function <em>T</em>(<em>m</em>,<em>n</em>)</p> by <em>T(m</em>,1) = <em>m</em> and <em>T</em>(<em>m</em>,<em>k</em> + 1) = <em>m</em><sup><em>T</em>(<em>m</em>,<em>k</em>)</sup>. We prove that if <em>G</em> is a finite <em>m</em>-generator group of exponent 8 then |<em>G</em>| ≤ <em>T(m</em>, 7<sup>471</sup>), improving upon the best previously known bound of <em>T(m</em>, 8<sup>8<sup>8</sup></sup>). |
| spellingShingle | Lie algebras Lie groups Groves, D Daniel Groves Problems in Lie rings and groups |
| title | Problems in Lie rings and groups |
| title_full | Problems in Lie rings and groups |
| title_fullStr | Problems in Lie rings and groups |
| title_full_unstemmed | Problems in Lie rings and groups |
| title_short | Problems in Lie rings and groups |
| title_sort | problems in lie rings and groups |
| topic | Lie algebras Lie groups |
| work_keys_str_mv | AT grovesd problemsinlieringsandgroups AT danielgroves problemsinlieringsandgroups |