多重循环群的一个注记  

A Note on Polycyclic Groups

作　　者：刘合国 张继平[2] 徐行忠 廖军[2] He Guo LIU;Ji Ping ZHANG;Xing Zhong;XU Jun LIAO(Department of Mathematics,Hainan University,Haikou 570228,P.R.China;School of Mathematical Sciences,Peking University,Beijing 100871,P.R.China;School of Mathematics and Statistics,Hubei University,Wuhan 430062,P.R.China)

机构地区：[1]海南大学数学系,海口570228 [2]北京大学数学科学学院,北京100871 [3]湖北大学数学与统计学学院,武汉430062

出　　处：《数学学报（中文版）》2023年第3期399-404,共6页Acta Mathematica Sinica：Chinese Series

基　　金：国家自然科学基金资助项目(11131001,11971155,12071117,12171142)。

摘　　要：设A是秩为n的自由Abel群.熟知A的自同构群Aut(A)=GL(n,Z).设f(λ)=λ^(n)+a_(n-1)λ^(n-1)+…+a_(1)λ+a_(0)∈Z[λ]是不可约多项式,其中a0=±1.设T=<α>是无限循环群,α通过多项式f(λ)的Frobenius相伴矩阵诱导的自同构作用在A上.设G=A■T.我们证明G是剩余有限p-群当且仅当p整除f(1).Let A be a free abelian group of rank n.It is well known that the automorphism group Aut(A)of A is GL(n,z).f(λ)=λ^(n)+a_(n-1)λ^(n-1)+…+a_(1)λ+a_(0) be an irreducible polynomial in Z[λ],where a_(0)=±1.Let T=<α>be an infinite cyclic group.Letαact on A via the automorphism of A induced by the Frobenius companion matrix of the monic polynomial f(λ).Assume that G=A■T.Let p be a prime.We prove that G is a residually-finite p-group if and only if p divides f(1).

关 键 词：剩余有限p-群 多重循环群 不可约多项式 

分 类 号：O152[理学—数学]