No abstract

G. Navarro has conjectured a necessary and sufficient condition for a finite group G to have a self-normalizing Sylow 2-subgroup, which is given in terms of the ordinary irreducible characters of G. In a previous article, the author has reduced the proof of this conjecture to showing that certain related statements hold for simple groups. In this article, we describe the action of Galois automorphisms on the Howlett-Lehrer parametrization of Harish-Chandra induced characters. We use this to complete the proof of the conjecture by showing that the remaining simple groups satisfy the required conditions. Mathematics Classification Number: 20C15, 20C33 Lemma 2.1. Let M be an A-module affording the character η of A and let B be a basis for M . Given a ∈ A, we have [ρ σ (a σ )] B σ = [ρ(a)] σ B ∈ Mat dim M (F) and η σ (a σ ) = η(a) σ .

The so-called "local-global" conjectures in the representation theory of finite groups relate the representation theory of G to that of certain proper subgroups, such as the normalizers of particular p-groups. Recent results by several authors reduce some of these conjectures to showing that a certain collection of stronger conditions holds for all finite simple groups. Here, we show that G = Sp 6 (2 a ) is "good" for these reductions for the McKay conjecture, the Alperin weight conjecture, and their blockwise versions.

We prove a variation of Thompson's Theorem. Namely, if the first column of the character table of a finite group G contains only two distinct values not divisible by a given prime number p ą 3, then O pp 1 pp 1 pGq " 1. This is done by using the classification of finite simple groups. Mathematics Classification Number: 20C15, 20C33

The Alperin-McKay conjecture relates height zero characters of an -block with the ones of its Brauer correspondent. This conjecture has been reduced to the so-called inductive Alperin-McKay conditions about quasi-simple groups by the third author. The validity of those conditions is still open for groups of Lie type. The present paper describes characters of height zero in -blocks of groups of Lie type over a field with q elements when divides q − 1. We also give information about -blocks and Brauer correspondents. As an application we show that quasi-simple groups of type C over F q satisfy the inductive Alperin-McKay conditions for primes ≥ 5 and dividing q − 1. Some methods to that end are adapted from [MS16].