Title:

Width questions for finite simple groups

This thesis is concerned with the study of nonabelian finite simple groups and their generation. In particular, we are interested in a class of problems called width questions which measure the rate at which a given collection of elements generates a group. Let G be a finite group and let S ⊂ G be a subset that generates G. Then the width of G with respect to S is defined to be the minimal k ∈ N such that any element of G can be written as a product of at most k elements from S. The focus of this work is the width of nonabelian finite simple groups with respect to the set of elements of order p, where p is a fixed prime. We call this the pwidth of the group G. The work of Liebeck and Shalev shows that there exists an absolute constant N > 0 that bounds the pwidth of any finite simple group (of order divisible by p), and in this thesis we seek the minimal value of N. Of particular interest is the case where p = 2, as the study of involutions plays a large role in the overall study of finite simple groups. The involution width of finite simple groups has received considerable attention in the literature: notably there exists a classification of finite simple groups of involution width two (the socalled strongly real groups). The first main result of this thesis completes the involution width problem: we show that every nonabelian finite simple group has involution width at most four. Furthermore, this result is sharp, as there exist families with involution width precisely four. The proof of this result makes extensive use of the representation theory of finite groups of Lie type, and in particular we develop the theory of minimal degree characters using dual pairs. In the latter part of the thesis, we extend our methods to consider the pwidth for odd primes. We partially resolve this problem, obtaining sharp bounds for the pwidth of alternating groups and sporadic groups: for any odd prime p, the pwidth of An (n ≥ p) is at most three, whereas the pwidth of a sporadic group is two, except for a small number of known exceptions. We also consider the pwidth for some groups of Lie type of small rank.
