A Characterization of Finite Simple Groups by the Degrees of Vertices of Their Prime Graphs

• 来源期刊: 代数集刊(英文版)
• 关键词: simple group prime graph degree of a vertex degree pattern order component
• 年，卷（期）: 2005,（3）
• 页码范围: 431-442
• 页数: 12页
• 语种: 英文

摘要

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 there is an element in G of order pq.Assume|G|=p1α1p2α2…pκακ with primes p1<p2<…pκ and natural numbers αi.For p∈π(G),let the degree of p be deg (q)=|{q∈π(G)|q～p}|,and D(G):=(deg(p1),deg(p2),…,deg(pκ)),in this paper,we prove that if G is a finite group such that D(G) = D(M) and |G| = |M|, where M is one of the following simple groups: (1) sporadic simple groups, (2) alternating groups Ap with p and p - 2 primes, (3) some simple groups of Lie type, then G ≌ M. Moreover,we show that if G is a finite group with OC(G) = {29.39.5.7, 13}, then G ≌ S6(3) or O7(3), and finally, we show that if G is a finite group such that |G| = 29.39.5.7.13 and D(G) = (3, 2, 2, 1, 0), then G ≌ S6(3) or O7(3).