L15(2)的新刻画  

New Characterizations of L_(15)(2)

作　　者：张良才[1] 张苗[1] 聂文敏[1] 

机构地区：[1]重庆大学数学与统计学院重庆沙坪坝401331

出　　处：《数学物理学报（A辑）》2013年第6期1169-1177,共9页Acta Mathematica Scientia

基　　金：国家自然科学基金(11271301,11171364,10871032);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)={p_1,p_2,…,p_s}.对于任意p∈π(G),令deg(p):=|{q∈π(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(p_1),deg(p_2),…,deg(p_s)),其中p_1<p_2<…<p_s,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L_(15)(2)是可OD-刻画的.作为一个推论,得到L_(15)(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.If G is a finite group, we define its prime graph F（G） as follows. The vertices of F（G） are the primes dividing the order of G and two distinct vertices p, q are joined by an edge, denoted by p N q, if and only if there is an element in G of order pq （see [7, 15]）. Ozk Assume IGI = plp2...pk with primes pl 〈 P2 〈 〈 Pk and natural numbers ai. For p {Pl,Pa, ,Pk}, define deg（p） ：= [{q e 7r（G）[q p}[, which is called the degree of p. We also define D（G） ：= （deg（pl）, 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 IMI = IGI 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-eharacterizable 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-eharaeterizable.

关 键 词：有限单群 素图 顶点度数 素图度数序列 

分 类 号：O152.1[理学—数学]