Title:

Some combinatorial problems in group theory

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.
