有限群的非p-幂零子群的正规化子

It is very important to investigate the relationship between automorphism groups of finite groups and the structure of finite groups in the theory of finite groups. In this field, the power automorphism is one of the most interesting objects. In 1934, Baer introduced a concept of norm in finite groups, that is the intesection of the normalizers of all subgroups of a finite group G. The importance lies in that every element of norm of a finite group can induce a power automorphism of the group and many interesting results has been given. Recently,``generalized norms” become popular and some new problems have been raised.. In this project, we first introduce the relationship between the structure of a finite group G and the intersection of the normalizers of all non p-nilpotent subgroups of G; Next, we study the relationship between the intersection of the normalizers of all non p-nilpotent subgroups and the intersection of the normalizers of all non-nilpotent subgroups in a finite group G; Finally, we give the relationship between the intersection of the normalizers of p-nilpotent residuals of all subgroups and the intersection of the normalizers of all non p-nilpotent subgroups in a finite group G.

有限群的自同构群与有限群的结构之间的关系是有限群论研究中的一个重要课题。在有限群的自同构中，群的幂自同构是人们非常感兴趣的研究对象之一。 Norm在研究群的幂自同构中起着非常重要的作用。1934年，Baer引入norm的概念(群G的所有子群的正规化子的交)，其重要性在于有限群的norm中的每个元素诱导该群的一个幂自同构。从而吸引了许多群论工作者从事这一领域的研究，并获得了大量的研究成果。随着研究的深入，群论学者提出了“广义norm”的概念并引起人们的极大关注。.本项目中，我们首先考虑有限群G的所有非p-幂零子群的正规化子的交对群结构的影响；其次，研究有限群G的所有非p-幂零子群的正规化子的交与G的所有非幂零子群的正规化子的交之间的关系；最后，考虑有限群G的所有子群的p-幂零剩余的正规化子的交与所有非p-幂零子群的正规化子的交之间的关系。