Müyesser, Alp;
(2024)
Large-scale structures in groups.
Doctoral thesis (Ph.D), UCL (University College London).
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Abstract
This thesis is broadly concerned with finding perfect matchings in hypergraphs whose vertices represent group elements and edges represent solutions to systems of linear equations. For example, given a subset of a group, when is it possible to partition the subset into triples whose products are the identity element? A well-known problem in this direction is the Hall-Paige conjecture from 1955 which asks for a characterisation of all groups whose multiplication table (viewed as a Latin square) contains a transversal. Many problems in the area have a similar flavour, yet until recently they have been approached in completely different ways, using mostly algebraic tools ranging from the combinatorial Nullstellensatz to Fourier analysis. The main result in this thesis gives a unified approach to attack these problems, using tools from probabilistic combinatorics. In particular, we derive that a suitably randomised version of the Hall-Paige conjecture can be used as a black-box to settle many old problems in the area for sufficiently large groups. As a by-product, we obtain the first combinatorial proof of the Hall-Paige conjecture. The second result in this thesis refines these tools further to solve a problem concerning the existence of transversals with a prescribed cycle type, confirming a conjecture of Friedlander, Gordon, and Tannenbaum from 1981.
| Type: | Thesis (Doctoral) |
|---|---|
| Qualification: | Ph.D |
| Title: | Large-scale structures in groups |
| Open access status: | An open access version is available from UCL Discovery |
| Language: | English |
| Additional information: | Copyright © The Author 2024. Original content in this thesis is licensed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) Licence (https://creativecommons.org/licenses/by-nc/4.0/). Any third-party copyright material present remains the property of its respective owner(s) and is licensed under its existing terms. Access may initially be restricted at the author’s request. |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10194469 |
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