Character Of Some Frobenius Groups And Non-commutative Simple Group Less Than 10 000 Order Character Block

Character theory plays an important role in the development of group theory, especially in the investigation of the structure of a finite group.In this paper,we investigate the existence and the character tables of Frobenius groups of order p^mq^αwith cyclic Frobenius component and cyclic Frobenius kernel. We get Theorem 1.1 and Table 1.1.We get Theorem 2.1,Theorem 2.2,Theorem 2.3,Theorem 2.4,Theorem 2.5 by using group theory,number theory and character theory together.Use these theorems,we study the separating characters by blocks of groups in the following types:(1) Frobenius groups of order p^mq^αwith cyclic Frobenius component and cyclic Frobenius kernel;(2) D_2n(n≥3,2(?)n);(3) groups of direct product.We know that a finite abelian group is a simple group,if and only if its order is a prime.The character table and the character blocks of it are all well known. In the last section,we use Theorem 2.1,Theorem 2.2,Theorem 2.3 to separate the irreducible characters of simple groups whose orders are less than 10000(except group G = L₂(16)).We still give some information about whether or not I_rr(G) isπ-separated.Because there is an element y₁₅ in the character table of group G = L₂(16),and we still don't know the property of it completely,so we just give out the result of its separating and conclude some property of y₁₅.