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Title: Some combinatorial problems in group theory
Author: Eberhard, Sean
ISNI:       0000 0004 6495 9579
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2016
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We study a number of problems of a group-theoretic origin or nature, but from a strongly additive-combinatorial or analytic perspective. Specifically, we consider the following particular problems. 1. Given an arbitrary set of n positive integers, how large a subset can you be sure to find which is sum-free, i.e., which contains no two elements x and y as well as their sum x+y? More generally, given a linear homogeneous equation E, how large a subset can you be sure to find which contains no solutions to E? 2. Given a finite group G, suppose we measure the degree of abelianness of G by its commuting probability Pr(G), i.e., the proportion of pairs of elements x,y Ε G which commute. What are the possible values of Pr(G)? What is the set of all possible values like as a subset of [0,1]? 3. What is the probability that a random permutation π Ε Sn has a fixed set of some predetermined size k? Particularly, how does this probability change as k grows? We give satisfactory answers to each of these questions, using a range of methods. More detailed abstracts are included at the beginning of each chapter.
Supervisor: Green, Ben Sponsor: Mathematical Institute ; University of Oxford
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available