This thesis is concerned with the study of non-abelian 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 non-abelian finite simple groups with respect to the set of elements of order p, where p is a fixed prime. We call this the p-width of the group G. The work of Liebeck and Shalev shows that there exists an absolute constant N > 0 that bounds the p-width 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 so-called strongly real groups). The first main result of this thesis completes the involution width problem: we show that every non-abelian 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 p-width for odd primes. We partially resolve this problem, obtaining sharp bounds for the p-width of alternating groups and sporadic groups: for any odd prime p, the p-width of An (n ≥ p) is at most three, whereas the p-width of a sporadic group is two, except for a small number of known exceptions. We also consider the p-width for some groups of Lie type of small rank.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:724199 |
| Date | January 2017 |
| Creators | Malcolm, Alexander |
| Contributors | Liebeck, Martin |
| Publisher | Imperial College London |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://hdl.handle.net/10044/1/50703 |
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