Some Properties Of Fully Graded Algebras And Character Rings Of Finite Groups

This dissertation focuses on the fully graded algebras over a complete discrete valuation ring with an algebraically closed residue field of characteristic p; the number of the connected components of the prime spectrum of some kinds of extension rings of the character rings of finite groups; and the generalized Brauer induction theorem. It is composed of four parts. In the preface, we first introduce the background and main ideas of the present research, and then we list all of our main results of each chapter. In the first chapter, we discuss the fully graded algebras over discrete valuation ring with an algebraically closed residue field of characteristic p. Firstly, we obtain a method to decompose the identity element such that the length of the decomposition is bounded by a constant; secondly, we get the so-called Fong theorem for the fully graded algebras over finite p-solvable groups; thirdly, we study the construction of some special kinds of fully graded algebras over commutative local rings. We describe the connected components of the prime spectrum of some kinds of extension rings of the character rings of some special kinds of finite group in Chapter 2. The last chapter is concerned with the induction theorem for the extension rings of the character rings of finite groups.Let G be a finite group. In the first chapter, we consider the fully G-graded algebras over a complete valuation ring with an algebraically closed residue field of characteristic p. In the first place, we prove that when the 1-component A₁ of the fully G- graded algebra A is isotypic , A is a crossed product of G over A\. By using this lemma , we reach a way to decompose the identity element of the fully G-graded algebra at any x ∈ G with the length bounded by a constant; for a finite p-solvable group G, we get the Fong theorem for the fully G-graded algebra A over a complete valuation ring with an algebraically closed residue field of characteristic p. The theorem tells us that for any Hall p'-subgroup H of the finite p-solvable group G, any primitive idempotent in A is conjugate to an primitive idempotent in the H-part A_H of A; we also show a necessary and sufficient condition for a primitive idempotentof AH still being primitive in A. In the second place, let O be a commutative local ring. We think about the fully G-graded algebras B with 1-component isomorphic to a direct sum of full matrix algebras over O ; here G has a natural action on the indecomposable direct summand of B\. We prove that when the action is regular, B itself is isomorphic to a full matrix algebras over O; and that when the action is semi-regular, B is isomorphic to a direct sum of some full matrix algebrs over O.Let G be a finite group with exponent ea and w be a primitive e^-th root of unity. Suppose S is a subring of the algebraic number field containing the rational integer number ring Z, and n is a set of prime numbers defined below:7T = {p| p is a prime number such that p^* ï¿¡ S}.Assuming further that G has a normal Hall-7r subgroup, denoting the character ring of G by R(G), in chapter 2 we prove that the number of connected components in Spec(5 z R(G)) equals the number of it -regular conjugacy classes in G.In the third chapter, G denotes a finite group and S expresses a subring of the complex field containing the ration integer number ring as subring. it is a set of prime numbers defined as follows:7T = {p\ p is a prime number such that p^* ï¿¡ S[uj}}.where a; is a primitive e^-th root of unity, ea is the exponent of G and S[uj] is the extended ring of S generated by u. We still denote R{G) the character ring of G. We show that the minimal family of subgroups Ys of G such that the following mapS ?z Ind : ?HeYs Sis a surjection is equal to W(tt); W(tt) is the union of all the elementary p-subgroups, where p runs over all the prime numbers in w (when tt is an empty set, we set W(tx) be the family of all the cyclic subgroups of G).