| 刘合国,吴佐慧,张继平,徐行忠,廖军.无限循环群被有限生成Abel群的中心扩张[J].数学年刊A辑,2015,36(3):233~246 |
| 无限循环群被有限生成Abel群的中心扩张 |
| On Central Extensions of Infinite Cyclic Groups byFinitely Generated Abelian Groups |
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| DOI： |
| 中文关键词: 中心扩张, 有限生成Abel群, 中心, 换位子群, 不变量 |
| 英文关键词:Central extension, Finitely generated Abelian group,
Center, Commutator subgroup, Invariant |
| 基金项目:本文受到湖北省高层次人才工程基金(No.1070-016533)和国家自然科学基金 (No.11131001, No.1137 1124, No.11401186)的资助. |
| Author Name | Affiliation | E-mail | | LIU Heguo | School of Mathematics and Statistics, Hubei University, Wuhan 430062, China. | ghliu@hubu.edu.cn | | WU Zuohui | School of Mathematics and Statistics, Hubei University, Wuhan 430062, China. | zuohuiwoo@163.com | | ZHANG Jiping | School of Mathematical Sciences, Peking University, Beijing 100871, China. | jzhang@math.pku.edu.cn | | XU Xingzhong | School of Mathematics and Statistics, Hubei University, Wuhan 430062, China. | xuxingzhong407@126.com | | LIAO Jun | Corresponding author. School of Mathematics and Statistics, Hubei University,Wuhan 430062, China. | jliao@hubu.edu.cn |
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| 中文摘要: |
| 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$\zeta G$的挠子群.
如果$T$的阶与$\zeta G/(G'\oplus T)$的挠子群的阶互素, 那么
群$G$可分解为$G=S\times F\times T$, 其中
$$
S=\left\{\left(
\begin{array}{cccccc}
1&d_1\alpha_{1}&d_2\alpha_{2}&\cdots&d_r\alpha_{r}&\alpha_{r+1}\0&1&0&\cdots&0&\alpha_{r+2}\\vdots&\vdots&\vdots& &\vdots&\vdots\0&0&0&\cdots&0&\alpha_{2r}\0&0&0&\cdots&1&\alpha_{2r+1}\0&0&0&\cdots&0&1
\end{array}
\right)\left|
\begin{aligned}
\\\alpha_{j}\in \mathbb{Z} \\~\ \end{aligned}
\right.
\right\},
$$
这里$d_i$都是正整数, 满足$d_1\mid d_2\mid \cdots \mid d_r$, $F$是秩为$s$的自由Abel群,
$T$是有限Abel群, $T=\mathbb{Z}_{e_1}\oplus \mathbb{Z}_{e_2}\oplus\cdots\oplus\mathbb{Z}_{e_t}$, $e_1>1$,
满足$e_1\mid e_2\mid \cdots \mid e_t$, 并且$(d_1, e_t)=1$.
进一步, $(d_1, d_2,\cdots , d_r; s;e_1,e_2,\cdots , e_t)$ 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$\zeta H$的挠子群.
如果$T_{H}$的阶与$\zeta H/(H'\oplus T_{H})$的挠子群的阶互素,
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. |
| 英文摘要: |
| 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 G of G. If the order of
T is prime to the order of the torsion subgroup of G/(G′ ⊕T ), then G has a decomposition
G = S × F × T , where
S =
??????????
?????????
?
????????
1 d1 1 d2 2 · · · dr r r+1
0 1 0 · · · 0 r+2
...
...
...
...
...
0 0 0 · · · 0 2r
0 0 0 · · · 1 2 r+1
0 0 0 · · · 0 1
?
????????
j ∈ Z
??????????
?????????
,
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 = Ze1 ⊕ Ze2 ⊕ · · · ⊕ Zetwith e1 > 1, e1 | e2 | · · · | et,
and (d1, et) = 1. Moreover, (d1, d2, · · · , dr; 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 H is prime to that
of the torsion subgroup of H/(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 |
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