L_7(3)与GL_7(3)的OD-刻画

OD-Characterization of L_7(3) and GL_7(3)

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构成的集合.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,g相连(记作p～q)的充要条件是群G恰有pq阶元.设π(G)={P_1,p2,…,p_x}.对于任意给定的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,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.引入了一个新的引理并证明了特殊射影线性群L7(3)是可OD-刻画的;一般线性群GL7(3)是可3-重OD-刻画的.作为一个推论,得到L7(3)是可OG-刻画的.

If G is a finite group, we denote by π（G） the set of all the prime factors of the order of G and construct a simple graph F（G）, called the prime graph of G, as follows. The vertices of F（G） are π（G） and two distinct vertices p, q are joined by an edge, denoted by p -q, if and only if there is an element of order pq in G. For any given p E∈（G） = （Pl,P2, ... ,Ps）, define degdeg（p）：=｜{q∈π（G）｜P in r（G）}l, which is called the degree of p. Define D（G） =（deg（pl）, deg（p2）,... , deg（p8））, where Pl 〈 P2 〈 ... 〈 Ps, called the degree pattern of the prime graph of G. A group G is k-fold OD-characterizable if there exist exactly k nonisomorphic finite groups M with the same order and degree pattern as M. In particular, a 1-fold OD-characterizable group is simply called an OD-characterizable group. In the present paper, it is proved that the projective linear group L7（3） is OD-characterizable by a newly introduced lemma and the general linear group GLT（3） is 3-fold OD-characterizable. As a consequence, it is obtained that L7（3） is OG-characterizable.