关于Seksenbaev-Robinson定理  

On Seksenbaev-Robinson Theorem

作　　者：刘合国 张继平[2] 赵静[1,3] 徐行忠 廖军 LIU Heguo;ZHANG Jiping;ZHAO Jing;XU Xingzhong;LIAO Jun(School of Mathematics and Statistics,Hainan University,Haikou 570228,China;School of Mathematical Sciences,Peking University,Beijing 100871,China;Key Laboratory of Engineering Modeling and Statistical Computation of Hainan Province,Hainan University,Haikou 570228,China;School of Mathematics and Statistics,Hubei University,Wuhan 430062,China)

机构地区：[1]海南大学数学与统计学院,海口570228 [2]北京大学数学科学学院,北京100871 [3]海南省工程建模与统计计算重点实验室,海口570228 [4]湖北大学数学与统计学学院,武汉430062

出　　处：《数学年刊（A辑）》2024年第2期185-204,共20页Chinese Annals of Mathematics

基　　金：国家自然科学基金(No.11131001,No.11971155,No.12071117,No.12171142);湖北省自然科学基金(No.2021CFB479)的资助。

摘　　要：剩余有限群也被称为是可以有限逼近的群,其特性常常由它的有限商群的性质决定.Seksenbaev定理断言:若对无限多个素数p,多重循环群G都是剩余有限p-群,则G是有限生成的无挠幂零群.Robinson把该定理推广为:设G是有限秩的可解群,若对无限多个素数p,G都是剩余有限p-群,则G是有限秩的无挠幂零群.这是无限可解群里的两个经典结果.本文证明了有关无限可解群的两个剩余有限性定理.本文的结果完善了Seksenbaev-Robinson定理.The residually finite group is also known as finitely approximable group,and its properties are often determined by the properties of its finite quotient groups.The Seksenbaev theorem states that if a polycyclic group G is a residually finite p-group for infinitely many primes p,then G is a finitely generated torsion-free nilpotent group.Robinson generalizes this theorem to solvable groups:If a solvable group G of finite rank is a residually finite p-group for infinitely many primes p,then it is a torsion-free nilpotent group.These are two classical results in infinite solvable groups.This paper proves two residual finiteness theorems for infinite solvable groups.The results of this paper improve the Seksenbaev-Robinson theorem.

关 键 词：可解群 剩余有限性 不可约多项式 整群环 下中心列 

分 类 号：O152[理学—数学]