有限群的幂自同构与子群格  

The Power Automorphisms and the Subgroup Lattices in Finite Groups

作　　者：郭秀云 王俊新[3] GUO Xiuyun;WANG Junxin(College of Mathematics and Systems Science,Shandong University of Science and Technology,Qingdao,Shandong,266590,P.R.China;Department of Mathematics,Shanghai University,Shanghai,200444,P.R.China;School of Applied Mathematics,Shanxi University of Finance and Economics,Taiyuan,Shanci,030006,P.R.China)

机构地区：[1]山东科技大学数学与系统科学学院,山东青岛266590 [2]上海大学数学系,上海200444 [3]山西财经大学应用数学学院,山西太原030006

出　　处：《数学进展》2025年第2期357-378,共22页Advances in Mathematics(China)

基　　金：国家自然科学基金(No.12171302)。

摘　　要：群与格是现代数学中两个最基本的研究对象.本文主要介绍群的幂自同构与群的子群格之间的相互关联,以及近年来围绕群的幂自同构与群的子群格方面人们所取得的一些新的研究成果.我们将从有限群的幂自同构与有限群的结构,有限群的norm、Wielandt-子群以及Wielandt-子群链与有限群的结构,有限群的子群格与有限群的结构,有限群的二极大子群、弱二极大子群与有限群的结构等四个方面来阐述.同时,一些可以进一步研究的问题被提出.Groups and lattices are the two most fundamental research objects in modern mathematics.In this paper,we mainly introduce the relationship between power automorphisms of groups and subgroup lattices of groups,and the new research findings in power automorphisms and subgroup lattices made in recent years.We will elaborate in four areas:the power automorphisms of finite groups and the structure of finite groups;norm,Wielandt subgroups,Wielandt subgroup chains and the structure of finite groups;the subgroup lattices of finite groups and the structure of finite groups;second maximal subgroups,weak second maximal subgroups and the structure of finite groups.Meanwhile,some problems that can be further investigated are put forward.

关 键 词：幂自同构 子群格 NORM Wielandt-子群链 极大子群 二极大子群 弱二极大子群 Pálfy-Pudlák问题 

分 类 号：O152.2[理学—数学]