| Abstract |
The representation theory of finite groups was created by Ferdinard Frobenius (1849-1917) in a great series of papers published in 1890s, which contained much of what is known today as the character theory of finite groups. Soon, William Burnside (1852-1927) developed his own approach to the representation theory, and applied it in the group theory. Of course, they were later joined by Issai Schur (1875-1941) who gave a new introduction to the representation theory based on elementary facts from linear algebra, that opened the subject to a wide audience, and later by Richard Brauer (1901-1977) who set the background for modular representation theory.
When character theory was developed, it was primarily viewed as a powerful tool for proving theorems about finite groups, for instance, Frobenius's theorem on what we now call Frobenius kernels. Other well known examples are Burnside's paqb - theorem and the Feit-Thompson odd-order theorem. In addition, character theory makes a central contribution to the complete classification of finite simple groups.
In character theory there is a number of theorems that have a common general form, the number of irreducible characters of a group G that satisfy a certain condition is equal to the number of irreducible characters of some related group H with the same condition. Often, there exists a bijection connecting the sets that is independent of the choices that may have been made in its construction. We call such a map, a canonical or natural correspondence. Best examples of this kind of correspondences are the Clifiord correspondence and the Gallagher correspondence.
Another, much deeper example is the Glauberman correspondence, which lies at the heart of the so-called global-local counting conjectures. The final example is the (still-unproved) McKay conjecture which is the origin of the global-local counting conjectures in the representation theory of finite groups. One could say that a new era began in the representation theory of finite groups with the formulation of the McKay conjecture.
In order to study the McKay conjecture, some work has to be done. In Chapter 1 we establish our basics from group theory. Main source for this chapter is [12]. Chapter 2 contains the basic material in character theory and in Chapter 3 we present some fundamental techniques on normal subgroups, such as Gallagher's theory on the extendability of characters and fully ramified characters. We also allow ourselves some digressions, in order to define an interesting family of groups, the extraspecial groups and finally we mention Brauer's theorems on the characterization and induction of characters. This chapter is based on [10] and [20].
Chapter 4 is divided in two sections both concentrated on the Glauberman correspondence. In the first section we present the construction of the Glauberman correspondence when the group acting is solvable. For the proof we follow G. Glauberman's approach as it is presented in [10]. In the second section, we follow G. Navarro's approach [20] to obtain the correspondence when the group acting is a p-group, by using some algebra homomorphisms. We also consider some questions (based on a expository paper of Navarro [23]) related to the Glauberman correspondence.
In Chapter 5, we focus on the McKay conjecture and apply the Glauberman correspondence to obtain the conjecture when the group has a normal p-complement. Finally, we give a detailed description of T. R. Wolf's proof in the solvable case. In order to do this, it is necessary to consider fully ramified characters for solvable groups and character triple isomorphisms. Finally, we present some refinements of the McKay conjecture and some of its consequences in group theory.
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