Non-Abelian Composition Factors of m-Rational Groups

Abstract

In this thesis, we discuss several problems in the representation theory of finite groups of Lie type. In Chapter 2, we will give essential background material that will be useful for the entirety of the thesis. We will investigate the construction of groups of Lie type, as well as their representations. We will define the field of values of a character afforded by a representation, and state useful results concerning these fields. In Chapter 3, we examine Zsigmondy primes and their existence, a necessary ingredient in proving our main results. In Chapters 4 and 5, we describe our main results in the ordinary and modular cases, which we now summarize. A finite group G is said to be m-rational, for a fixed positive integer m, if [Q(x):Q]|m for any irreducible character ꭓϵIrr(G). In 1976, R. Gow studied the structure of solvable rational groups (i.e. m=1), and found that the possible composition factors of a solvable rational group are cyclic groups of prime order p ϵ {2,3,5}, [22]. Just over a decade later, W. Feit and G. Seitz classified the possible non-abelian composition factors of (non-solvable) rational groups, [14]. In 2008, J. Thompson found an upper bound of p≤13 for the order of the possible cyclic composition factors of an arbitrary rational group, and conjectured that the bound can be improved to p≤5, [46]. More recently, J. McKay posed the question of determining the structure of quadratic rational groups (i.e. m=2). J. Tent studied the cyclic composition factors of solvable quadratic rational groups in 2013, [45]. In Chapter 4, we answer McKay's question concerning non-abelian composition factors, and generalize our results to non-solvable m-rational groups. Modular character theory was founded by R. Brauer in the 1930's, and has been useful in proving historical results including the classification of finite simple groups. In Chapter 5, we prove the modular version of our results. Though our conclusions are similar to those found in the complex case, the methods for proving the results are typically much more complicated.