基于申请人近期的研究工作以及Mckay猜想的研究现状，本项目重点研究以下问题：1.研究有限李型单群的Sylow子群的正规化子的极大正规p'-子群在该单群自同构群作用下的不变性；2.研究块图没有三角的有限群；3.研究恰好只有两个非有理的非线性不可约特征标的有限单群；4.研究非有理的非线性不可约特征标度数都相同的有限单群；5.研究有限李型群的半单元的中心化子不连通时群论与表示论方面的性质以及对应的Lusztig序列在自同构群作用下的表现；6.研究E6、2E6型例外群是否满足约化的Mckay条件；7.研究至少有两个不同不可约Brauer特征标度数能被定义特征整除的李型单群；8.研究恰好只有一个不可约Brauer特征标度数能被给定素数整除的有限群。这些研究对于发展与完善有限李型单群群论及其表示论具有非常重要的理论与实际意义。

Based on the applicant's recent research work and the current situation of Mckay conjecture, the present project mainly focuses on the following problems: 1. the stability of the largest normal p’-subgroup of the normalizer of a Sylow p-subgroup of a finite simple group of Lie type under the conjugate action of its automorphism group; 2. finite groups whose block graphs have no triangle; 3. finite simple groups with exactly two non-rational non-linear irreducible characters; 4. finite simple groups all of whose non-rational non-linear irreducible characters have the same degree; 5. the group and representation theoretic properties of the centralizers of semisimple elements of a finite simple group and how its automorphism group acts on the irreducible characters of the corresponding Lusztig series under the disconnectedness assumption of those centralizers; 6. whether or not finite simple groups of exceptional type E6, 2E6 satisfy the inductive Mckay condition; 7. finite simple groups of Lie type with at least two irreducible p-Brauer characters whose degrees are divisible by the defining characteristic p; and 8. finite groups with exactly one irreducible p-Brauer character degree divible by a given prime. The project has theoretical and practical significance in both group and representation theoretic development of finite simple groups of Lie type.