Characters In π-Separable Groups And Finite Groups With Some Arithmatic Properties

In this paper,we research the π-Brauer character relative to a normal subgroup in π-separable groups and groups with some special arithmetic properties, the paper consists of three parts. The preface introduces the backbounds and status quo on our two topics, and summarizes some researched results.In chapter one, we first introduce the .B_π-character theory in finite π-separable groups and some properties on π-Fong characters. For a Hall π-subgroup in π-separable group and a given π-Fong character, we show that there always exists a normal subgroup, such that some irreducible constituent of this π-Fong character which restricts on the join of these two subgroups is a π-Fong character associated to this normal subgroup. In section four, we consider a normal π-subgroup in π-separable group, a normal set and a complex value class function space defined on G, using the B_π-character theory and character-triple isomorphism, and find a classical basis of this space such that it behaves as the set of irreducible Brauer characters, and satisfies the Fong-Swan property. In section six, we follow the classical case and define the so called π-projective character, considering the class function space which vanishes on non π-regular elements, we prove that the π-projective character defined above is even a basis of this space. Especially, when π is only a prime number, the basis of the second space is just as the Kulshammer-Robinson Z-basis. Furthermore, when the normal subgroup is the idendity group, the basis of the first space is just as the set of irreducible Brauer characters.In chapter two, we investigate the relation between the finite group structure and its local information. When the degrees of irreducible characters of a finite group are all the product of the power of two distinct prime numbers, we obtain the inner structure of the finite group which satisfies this arithmetic property; at the same time, we also show that if the group has this structure, then the degrees of characters must satisfy this arithmetic property. In section four, we give two examples to illustrate some conditions in our theoremis absolutely necessarily.