Character Theory And Representations Of Finite Groups Of Lie Type

Group is one of the most important mathematical concepts and tools in modern science and the representations theory of group is a basic tool in the research of group theory. There are two active research interests in character theory which is the core of complex representation theory of finite group:the arithmetic properties of characters, conjugacy classes and the influence of the characters on group structure. The main results are the following three parts.Part1is about the arithmetic complexity of character degrees and conjugacy class sizes. In part1, we consider the core problem of character theory:Huppert’s ρ—σ conjecture and relative problems. Firstly, we improve Casolo and Dolfi’s result about the ρ-σ conjecture:for finite group G,|ρ(G)|≤7σ(G), which is the best result before and our result is|ρ(G)|≤6σ(G)+1. Next we consider the Huppert’s ρ’p-σ’p conjecture of p-regular conjugacy classes. For solvable group, we reduce the ρ’p-σ’p problem to the case of nilpotent-metabelian groups and the problem is partially solved in two special cases. At last, we reduce the ρp-σp problem and ρ’p-σ’p problem to a problem in number theory. By further study on the problem, we prove if prime p is fixed, the problems can be solved.Part2and Part3are about the influence of the characters on group structure. In part2, we consider the influence of the degree multiplicity on group structure. Group G is said to be a Dκ-group if|Irr1(G)|—|cd1(G)|=κ, Berkovich et al classify the D0-groups and D1-groups. On this basis we completely classify the D2-groups:if G is a nonsolvable D2-group, then G is isomorphic to one of S5, A6, A7, A9, M22or A10; if G is a solvable D2-group, there are18types of finite groups.In part3, we consider the influence of the special characters on group struc-ture. Thompson’s p-nilpotent theorem is a very profound result in character theory, Berkovich and Kazarin study the Thompson group’s weaker condition:group G such that|Irr1(G,p)|=1. Based on it, we study nonsolvable group G such that|Irr1(G,p’)|=2(p is odd prime) and our result is:If G is a nonsolvable group such that|Irr1(G,p’)|=2and p is odd prime, then p=3, G≌PSL2(7) or exists a2-group N such that G/N≌PS L2(5).