换位子群是不可分Abel群的有限秩可除幂零群  

Radicable Nilpotent Groups of Finite Rank with Indecomposable Abelian Commutator Subgroups

作　　者：刘合国[1] 张继平[2] 廖军[1] LIU Heguo;ZHANG Jiping;LIAO Jun(Department of Mathematics,Hubei University,Wuhan 430062,China;School of Mathematical Sciences,Peking University,Beijing 100871,China;Corresponding author.Department of Mathematics,Hubei University,Wuhan 430062,China)

机构地区：[1]湖北大学数学系,武汉430062 [2]北京大学数学科学学院,北京100871

出　　处：《数学年刊（A辑）》2018年第3期287-296,共10页Chinese Annals of Mathematics

基　　金：国家自然科学基金(No.11131001;No.11371124;No.11401186)的资助

摘　　要：完整地确定了换位子群是不可分Abel群的有限秩可除幂零群的结构,证明了下面的定理.设G是有限秩的可除幂零群,则G的换位子群是不可分Abel群当且仅当G'=Q或Q_p/Z且G可以分解为G=S×D,其中当G'=Q时,■当G'=Q_p/Z时,S有中心积分解S=S_1*S_2*…*S_r,并且可以将S形式化地写成■其中■,式中s,t都是非负整数,Q是有理数加群,π_κ(k=1,2,…,t)是某些素数的集合,满足π_1■Cπ_2■…■π_t,Q_π_k={m/n|(m,n)=1,m∈Z,n为正的π_k-数}.进一步地,当G'=Q时,(r;s;π_1,π_2,…,π_t)是群G的同构不变量;当G'=Q_p/Z时,(p,r;s;π_1,π_2,…,πt)是群G的同构不变量.即若群H也是有限秩的可除幂零群,它的换位子群是不可分Abel群,那么G同构于H的充分必要条件是它们有相同的不变量.The structure of the radicable nilpotent groups of finite rank with indecomposable abelian commutator subgroups is completely determined. More exactly, the following theorem is proved.Let G be a radicable nilpotent group of finite rank. Then the commutator subgroup of G is indecomposable and abelian if and only if G＇=Q or Q_p/Z and G has a decomposition G = S × D, where ■if G＇ = Q and S = S_1＊S_2 ＊…＊S_r,S_i≌S_p if G＇ = Q_p/Z. Write S formally as ■Here D is a divisible abelian group such that ■,where s and t are nonnegative integers, and Q is the additive group of the rational number field,where π_k（k=1,2,…,t） are the sets of some prime numbers such that π_1■C π_2■…≤π_t,and Q_π_k={m/n｜（m, n） = 1, m∈Z, nis a positive π_k-number}. Moreover,（p,r; s; π_1,π_2, …,π_t） is an isomorphic invariant of G, that is to say, if H is also a radicable nilpotent group of finite rank with indecomposable abelian commutator subgroup, then G is isomorphic to H if and only if they have the same invariants.

关 键 词：幂零群 局部循环群 中心 换位子群 可除群 

分 类 号：O152[理学—数学]