有限群的子群格与广幂自同构

格与群是现代数学中两个最基本的研究对象. 自从人们证明了每一个格都同构于某个群的子群格的子格之后，格便与群紧密的联系在一起. 而著名的Palfy–Pudlak 问题则正是有限格与有限群紧密联系的进一步体现， 也是有限群论和格论中人们非常感兴趣的一个公开未解决的问题. 本项目正是以研究著名的Palfy–Pudlak 问题为出发点，应用有限p-群的p-自同构的技巧，以及有限p’-群在有限p-群上的作用, 建立有限群的准无不动点自同构的中心化子与有限群结构之间的深刻关联. 进一步应用Bear和 Wielandt的思想，通过有限群的自同构群和幂自同构群来建立有限群的子群格与其上的自同构群之间的联系，从而给出有限群的几个群类使得其上的Palfy–Pudlak 问题成立. 相信我们的研究对有限群的自同构群的理论有推动作用，同时也为推动子群格的研究以及Palfy–Pudlak问题的解决有重要的理论意义.

Lattices and groups are two basic and important study subjects. Since it is proved that every lattice is isomorphic to a sublattice of the subgroup lattice for some group, it brings lattices and groups closer together. The famous Palfy-Pudlak problem is a further expression of the closed relationship between finite lattices and finite groups, and it is also an interesting open problem in finite group theory and lattice theory. The purpose of this project is to investigate the famous Palfy-Pudlak problem. We first set up the relationship between the centralizers of quasi- fixed-point-free automorphisms and the structure of finite groups by using “the skill of p-automorphisms of finite p-groups” erected by Khukhro and the action of finite p’-groups on finite p-groups. Next we set up the connection between the subgroup lattice of a finite group and its automorphism group by using the ideal of famous mathematicians Bear and Wielandt and the automorphism group (the power autpmorphism group) of the finite group. Finally, we may find some classes of finite groups such that Palfy-Pudlak problem is true on these classes. We believe that our investigation could make a positive role in promoting on the investigation of automorphism groups of finite groups, and it is an important theoretical significance for to study the subgroup lattices and Palfy-Pudlak problem.