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A New Characterization of Certain Symmetric and Alternating Groups

晏燕雄1,2,3, 陈贵云1, 徐海静1
YAN Yanxiong1, 2, 3, *, CHEN Guiyun1,**,XU Haijing1,***

1.西南大学数学与统计学院, 重庆, 400715;2. 重庆第二师范学院数学与信息工程系, 重庆, 400067; 3. 重庆师范大学数学科学学院, 重庆, 401331
1. School of Mathematics and Statistics, Southwest University, Chongqing, 400715, P. R. China; 2. Department of Mathematics and Information Engineering, Chongqing University of Education, Chongqing, 400067, P. R. China; 3. College of Mathematics Science,

收稿日期: 2011-08-05
出版日期: 2013-10-25
DOI: 10.11845/sxjz.20130503


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摘要 Algebra Colloquium}, 2005, 12(3): 431-442]提出与群G的素图有关的次数型D(G).群G称为k-重OD-刻画的, 如果恰好有 k 个不同构的群 M 使得 |G|=|M| 且 $D(G)=D(M)$. 并且 1-重 OD-刻画的群简称可 OD-刻画的. 以下单群能被其阶和次数型唯一决定: 散在单群, 交错群 $A_{p}$ (素数 $p\geq5$) 及某些李型单群. 关于群 G 的素图连通时对该问题的研究比较困难. 本文进行了这一研究, 证明了对称群 S81 和 S83 均是可 3-重 OD 刻画的. 另外, 本文也证明了交错群 A130 和 A140 是可 OD-刻画的, 该结果对文献[Frontiers of Mathematics in China}, 2009, 4(4): 669-680]提出的猜想给予了肯定的回答.
关键词 素图次数型顶点次数对称群交错群    
Abstract:The degree pattern of a finite group G has been introduced in [Algebra Colloquium}, 2005, 12(3): 431-442] and denoted by D(G). The group G is called k-fold OD-characterizable if there exist exactly k non-isomorphic groups $H$ satisfying conditions |G|=|M| and $D(H)=D(G)$. In addition, a 1-fold OD-characterizable group is simply called OD-characterizable. The following simple groups are uniquely determined by their orders and degree patterns: all sporadic simple groups, the alternating groups $A_{p}$ ($p\geq5$ is a twin prime) and some simple groups of Lie type. In this problem, those groups with connected prime graphs are somewhat much difficult to be solved. In this paper, we continue this investigation. In particular, we show that the symmetric groups S81 and S83 are 3-fold OD-characterizable. We also show that the alternating groups A130 and A140 are OD-characterizable. It is worth mentioning that the latter gives a positive answer to a conjecture in [Frontiers of Mathematics in China}, 2009, 4(4): 669-680].
Key wordsdegree pattern    degree of a vertex    symmetric group    alternating group
基金资助:Project partially supported by NSFC (No. 11171364, No. 11271301), the Natural Science Foundation Project of CQ CSTC (No. cstc2011jjA00020), the Fundamental Research Funds for the Central Universities (No. XDJK2009C074) and Graduate-Innovation Funds of Sci
[1] 赵红梅;唐国平;. 几类非Abel群之增广商群的结构(英文)[J]. 数学进展, 2008, 37(2): 163-170.
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