Some combinatorial problems in group theory

Thesis

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 ∈ 𝒮_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 ∈ 𝒮_n at random. It is well known that π₁,...,π_r will generate at least 𝒜_n with high probability as long as r ≥ 2, but what happens if we are allowed to replace π₁,...,π_r 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 𝒜_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.

Authors: Eberhard, S

• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford