Elements with large irreducible submodules contained in maximal subgroups of the general linear group

Pannek, Sabina Barbara; Hiß, Gerhard (Thesis advisor); Roney-Dougal, Colva M (Thesis advisor)

Aachen (2018, 2019)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2018. - Dissertation, University of Western Australia, 2018

Abstract

This thesis is concerned with a family of so-called fat elements in the finite general linear group GL(V) consisting of all non-singular linear mappings on a finite vector space V. We refer to an element of GL(V) as being fat if it leaves invariant, and acts irreducibly on, a subspace of dimension greater than dim(V)/2. Fatness of an element can be decided efficiently in practice by testing whether its characteristic polynomial f has an irreducible factor of degree greater than deg(f)/2.Fat elements generalise the concept of ppd-elements, which are defined by the property of having orders divisible by certain primes called primitive prime divisors. In 1997, Guralnick, Penttila, Praeger and Saxl classified all subgroups of GL(V) containing ppd-elements. Their work has had a wide variety of applications in computational group theory, number theory, permutation group theory, and geometry. Our overall goal is to carry out an analogous classification of all subgroups of GL(V) containing fat elements, and this thesis initiates that project. We first develop a comprehensive framework necessary for the study of fat elements. This includes new results in elementary number theory (concerning the order of an integer modulo r), theory of finite fields (counting certain irreducible polynomials) and group theory (regarding irreducible semilinear mappings). We then investigate the occurrence of fat elements in various subgroups of the general linear group GL(V). As in the case of the "ppd-classification", our analysis is patterned by Aschbacher's classification of the maximal subgroups of GL(V) into nine partly overlapping classes C1, ..., C8, and S. We investigate members of the Aschbacher classes C1, ..., C5, C7 and several representatives of the class S upon the existence of fat elements. Although a large majority of fat elements in GL(V) are ppd-elements, we show that members of certain Aschbacher classes with no (or hardly any) ppd-elements do contain fat elements. Therefore, the results we obtain in classes C2, C3, C4, C7 significantly differ from the findings of the "ppd-classification". For groups G contained in the Aschbacher classes C1, ..., C5 and (with restrictions in) C7 we additionally calculate the precise value of, and also determine good lower and upper bounds for, the proportion of all fat elements in G.