eprintid: 10194469
rev_number: 12
eprint_status: archive
userid: 699
dir: disk0/10/19/44/69
datestamp: 2024-09-25 14:11:12
lastmod: 2024-09-25 14:11:12
status_changed: 2024-09-25 14:11:12
type: thesis
metadata_visibility: show
sword_depositor: 699
creators_name: Müyesser, Alp
title: Large-scale structures in groups
ispublished: unpub
divisions: UCL
divisions: B04
divisions: C06
divisions: F59
note: 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.
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.
date: 2024-07-28
date_type: published
oa_status: green
full_text_type: other
thesis_class: doctoral_open
thesis_award: Ph.D
language: eng
primo: open
primo_central: open_green
verified: verified_manual
elements_id: 2295564
lyricists_name: Müyesser, Necati
lyricists_id: NAMUY86
actors_name: Müyesser, Necati
actors_id: NAMUY86
actors_role: owner
full_text_status: public
pages: 200
institution: UCL (University College London)
department: Mathematics
thesis_type: Doctoral
citation:        Müyesser, Alp;      (2024)    Large-scale structures in groups.                   Doctoral thesis  (Ph.D), UCL (University College London).     Green open access   
 
document_url: https://discovery.ucl.ac.uk/id/eprint/10194469/1/Phd_Thesis-6.pdf