Use this URL to cite or link to this record in EThOS:  https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.728742 
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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Abstract:  
We study a number of problems of a grouptheoretic origin or nature, but from a strongly additivecombinatorial 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 sumfree, 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:  uk.bl.ethos.728742  DOI:  Not available  
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