无限循环群被有限生成Abel群的中心扩张

On Central Extensions of Infinite Cyclic Groups by Finitely Generated Abelian Groups

设G是无限循环群被有限生成Abel群的中心扩张,T是G的中心ζG的挠子群.如果T的阶与ζG/(G'⊕T)的挠子群的阶互素,那么群G可分解为G=S×F×T,其中S= 这里d_i都是正整数,满足d_1|d_2|…|d_r,F是秩为s的自由Abel群,T是有限Abel群,T=Z_(e_1)⊕Z_(e_2)⊕…⊕Z_e_t,e_1>1,满足e_1|e_2|…|e_t,并且(d_1,e_t)=1.进一步,(d_1,d_2,…,d_T;s;e_1,e_2,…,e_t)是群G的同构不变量,即若群H也是无限循环群被有限生成Abel群的中心扩张,T_H是ζH的挠子群.如果T_H的阶与ζH/(H'⊕T_H)的挠子群的阶互索,那么G同构于H的充要条件是它们有相同的不变量.显然,这个结果涵盖了有限生成Abel群的结构定理.

Suppose that G is a central extension of an infinite cyclic group by a finitely generated Abelian group, and T is the torsion subgroup of the center ζG of G. If the order of T is prime to the order of the torsion subgroup of ζG/（G＇⊕T）, then G has a decomposition G = S × F × T, where di are positive integers satisfying d1｜d2｜…｜dr, F is a free Abelian group with rank s, and T is a finite Abelian group such that T=Z（e1）⊕Z（e2）⊕…⊕Zet,with e1 〉 1, e1｜e2｜…｜et, and （d1, et） = 1. Moreover, （d1,d2,…,dT;s;e1,e2,…,et） is an isomorphic invariant of G, that is to say, if H is also a central extension of an infinite cyclic group by a finitely generated Abelian group and the order of the torsion subgroup TH of ζH is prime to that of the torsion subgroup of ζH/（H＇⊕TH）, then G is isomorphic to H if and only if they have the same invariants. Obviously, this result covers the fundamental theorem for finitely generated Abelian groups.