数学物理学报 ›› 2013, Vol. 33 ›› Issue (6): 1169-1177.

• 论文 • 上一篇    下一篇

L15(2)}的新刻画

张良才*|张苗|聂文敏   

  1. 重庆大学 数学与统计学院 重庆 沙坪坝 |401331
  • 收稿日期:2012-07-13 修回日期:2013-02-07 出版日期:2013-12-25 发布日期:2013-12-25
  • 通讯作者: 张良才, E-mail:zlc213@163.com; 1073165848@qq.com; niewenmin1129@163.com
  • 基金资助:

    国家自然科学基金(11271301, 11171364, 10871032)和国家自然科学青年基金项目(11001226)资助

New Characterizations of L15(2)

 ZHANG Liang-Cai*, ZHANG Miao, NIE Wen-Min   

  1. College of Mathematics and Statistics, Chongqing University, Chongqing Shapingba |401331
  • Received:2012-07-13 Revised:2013-02-07 Online:2013-12-25 Published:2013-12-25
  • Contact: ZHANG Liang-Cai, E-mail:zlc213@163.com; 1073165848@qq.com; niewenmin1129@163.com
  • Supported by:

    国家自然科学基金(11271301, 11171364, 10871032)和国家自然科学青年基金项目(11001226)资助

摘要:

对于任意一个有限群G, 令π(G)表示由它的阶的所有素因子所构成的集合. 该文构建一种与之相关的简单图,称之为素图, 记
作Γ(G).该图的顶点集合是π(G),图中两顶点p, q相连(记作p~q)的充要条件是群G恰有pq阶元[7, 15]. 令π(G)={p1, p2,… , ps}.对
 于任意pπ(G),令deg(p):=|{qπ(G)|在素图Γ(G)中, p~q}|,并称之为顶点p 的度数. 同时, 我们定义D(G):=(deg(p1), deg(p2), …,dots(ps)), 其中p1<p2<…<ps, 并称之为群G的素图度数序列.若存在k个互不同构的群与群G具有相同的群阶和素图度数序列,则称群G是可k -重OD -刻画的. 特别地, 可1 -重OD -刻画的群也称为可OD -刻画的群[11]. 在该文中, 引入一个新的引理并证明了特殊射影线性群L15(2)是可OD -刻画的. 作为一个推论, 到L15(2)是可OG -刻画的. 该方法也可适用于其它一些具体的有限单群.

关键词: 有限单群, 素图, 顶点度数, 素图度数序列

Abstract:

If G is a finite group, we define its prime graph Γ(G) as follows. The vertices of Γ(G)  are the primes dividing the
order of G and two distinct vertices p, q are joined by an edge, denoted by p~q, if and only if there is an element in G of order pq (see [7, 15]). Assume |G|=p1α1p2α2pkαk with primes p1<p2<…<pk and natural numbers αi. For pπ(G)={p1, p2, … , pk, define deg(p):=|{qπ(G)|q~p}|, which is called the degree of p. We also define D(G):=(deg(p1), deg(p2), …, deg(pk)), which is called the degree pattern of the group G. We say a group G is t-fold OD-characterizable if there exist exactly t non-isomorphic finite groups M such that |M|=|G| and D(M)=D(G) (see [11]). In particular, a 1-fold OD-characterizable  group is simply called an OD-characterizable  group. In the present paper, we  prove  that the projective special linear group L15(2) is  OD-characterizable by a newly introduced lemma  to deal with its connected prime graph. As a consequence of this result, we obtain that  L15(2) is OG-characterizable.

Key words: Finite simple group, Prime graph, Degree of a vertex, Degree pattern

中图分类号: 

  • 20D05