Irreducible characters of Sylow p-Subgroups associated with some classical linear groups

Abstract

The classification of finite simple groups states that every finite simple group is isomorphic to one of the following groups: (1) a cyclic group with prime order, (2) an alternating group of degree ≥ 5, (3) a simple group of Lie type, including both the classical Lie groups, namely the simple groups related to the projective special linear, unitary, symplectic, or orthogonal transformations over a finite field, (4) the exceptional and twisted groups of Lie type (including the Tits group), (5) one of the 26 sporadic simple groups. More recent attention in group theory has been given to the important subgroups of these finite simple groups such as maximal subgroups and Sylow p−subgroups. We consider in this thesis some classical linear groups and in particular, we construct character tables of their Sylow p−subgroups. Since these Sylow p−subgroups sit in the maximal parabolic subgroups, we will also study specific maximal subgroups. A vast research exists on Sylow p − subgroups of classical linear groups and most of the motivation on the study has been driven by the conjecture asserting that the degrees of irreducible characters of Sylow p − subgroups of the general linear group are not merely powers of p, where p is the characteristic of the defined field, but powers of q, where q = p k is the order of the field. In fact, Martin Isaacs proved the conjecture in 1995 and further extended it to the symplectic group Sp2m(q) in which he established that the conjecture is false when p = 2. An even more interesting question regards the construction of the irreducible characters of these Sylow p − subgroups (note that even just counting them poses much difficulties). Since the groups considered in this thesis are of extension type, we will use the method developed by Bernd Fischer for the construction of character tables of group extensions. This method derives its rudiments from Clifford theory. In fact, as a result of this, most authors refer to the method as CliffordFischer matrices theory. Let G¯ = N.G, where N E G and G/N ¯ ∼= G be a group extension. For each representative of a conjugacy class of G, we form a matrix called the Fischer matrix on g. When all these Fischer matrices together with character tables, ordinary or projective, and fusions of inertia factor groups into G are obtained, one can then construct with ease the full character table of G¯. The first step to construct the character table of G¯ is to know its conjugacy classes. Jamshid Moori developed a technique to compute the conjugacy classes of group extensions. In this thesis, we apply the coset analysis technique together with the theory of Fischer matrices to construct the ordinary character tables of six groups of extension type, namely 26 :GL(3, 2), 26 :A8, 2 10:GL(4, 2), 25 :(2 × D8), 26 :(23 :D8) and 27 :(25 :(2 × D8)) associated with some classical linear groups. In addition, we give general form of Fischer matrices for a Sylow p − subgroup of Sp4(q) where q is a power of an odd prime p.