关于有限Abelian群的生成子集的基数

数学物理学报 ›› 2009, Vol. 29 ›› Issue (5): 1240-1245.

关于有限Abelian群的生成子集的基数

1.中南大学数学与计算技术学院长沙 410075|2.湖南第一师范学院数学系长沙 410002

收稿日期:2007-01-08 修回日期:2008-12-26 出版日期:2009-10-25 发布日期:2009-10-25
基金资助:
国家自然科学基金(10471152)资助

On the Cardinality of Generating Subsets of Finite Abelian Groups

1.School of Mathematics Sciences and Computing Technology, Central South University, Changsha 410075|2.Department of Mathematics, Hunan First Normal College, Changsha 410002)

Received:2007-01-08 Revised:2008-12-26 Online:2009-10-25 Published:2009-10-25
Supported by:
国家自然科学基金(10471152)资助

摘要/Abstract

摘要：

假若 G =Z_m₁Z_m₂… Z_mr 为 (m₁, m₂, …, m_r)型Abelian群, 其中Z_mi 为 m_i阶的循环群且1≤ i ≤ r, m₁|m₂|…| m_r, S 为G 的满足0 ∈ S=-S 的生成子集. 如果 |S|>|G|/ρ, 其中 ρ≥l m_r/2l 且m_r=e(G) 为群 G 的所有元素的阶的最小公倍数, 则ρS=G. 更进一步作者推广了Klopsch与lev ^[1]的一个结论,有:若 G=Z₂Z_m 为 (2, m) 型 Abelian 群(m ≥8), 则 t_m_/2(G)=0.

关键词: 基数, 生成集, 对称闭包

Abstract:

Suppose G =Z_m₁ Z_m_{2 ^…} Z_m_r be an Abelian group of type (m₁, m₂, …, m_r) (Z_mi is a cyclic group of order m_i, 1≤ i ≤ r, m₁|m₂| …| m_r). Let S be a symmetrically closed set (S is symmetrically closed if 0 ∈ S =-S) and a generating set of G. If |S|>|G|/ρ, where ρ ≥lm_r/2 land m_r=e(G) denotes the least common multiple of the orders of all elements of group G, then ρS=G. And if G=Z₂Z_m is an Abelian group of type (2, m) (m ≥ 8), then t _m/2}(G)=0, which extends the related results of Klopsch and Lev^[1].

Key words: Cardinality, Generating sets, Symmetric closure

中图分类号:

11D61

陈雪生, 胡亚辉, 唐明军. 关于有限Abelian群的生成子集的基数[J]. 数学物理学报, 2009, 29(5): 1240-1245.

CHEN Xue-Sheng, HU Ya-Hui, TANG Meng-Jun. On the Cardinality of Generating Subsets of Finite Abelian Groups[J]. Acta mathematica scientia,Series A, 2009, 29(5): 1240-1245.