Some combinatorial problems in group theory

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 ∈ 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 ∈ n has a fixed set of some predetermined size k? Particularly, how does this probability change as k grows? This problem is also related to the following one. Suppose we πck a few permutations π_1, ..., π_r ∈ n at random. It is well known that π1,...,πr will generate at least n with high probability as long as r ≥ 2, but what happens if we are allowed to replace π1,...,πr by arbitary conjugates π'1,..., π'r? 4. Pick two bijections π1,π2: {1,...,n} → Z/nZ uniformly at random. What is the probability that the pointwise sum π1 + π2 is also a bijection? This problem affords a fun interpretation ∈ terms of queens on a toroidal chessboard. 5. How big is the largest subset of the alternating group n which is product-free, i.e., which contains no two elements x and y as well as their product xy? 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.