交错群A5,A6及射影特殊线性群L2(7)的一种新刻画  

A New Characterization of the Alternating Group A,A and Projective Special Linear Group L(7)

作　　者：曹熠维 吕恒[1] CAO Yiwei;Lü Heng(School of Mathematics and Statistics,Southwest University,Chongqing 400715,China)

机构地区：[1]西南大学数学与统计学院,重庆400715

出　　处：《重庆师范大学学报（自然科学版）》2020年第4期96-99,共4页Journal of Chongqing Normal University:Natural Science

基　　金：国家自然科学基金(No.11971391)。

摘　　要：【目的】群的阶及群的共轭类的个数对群的结构有重要影响,探讨用群G的阶以及共轭类的个数k(G)刻画交错群A5,A6及射影特殊线性群L2(7)。【方法】利用有限群的共轭类的长与元素的中心化子的阶的关系及群的素图与群的结构的关系,使用分类讨论法及有限群的同构的判别法。【结果】由|G|=60,k(G)=5证明了G≌A5。同样地,由|G|=168,k(G)=6证明了G≌L2(7),由|G|=360,k(G)=7证明了G≌A6。【结论】通过给定群的阶以及共轭类的个数可以刻画某些有限单群,但是是否能够通过群的阶以及共轭类的个数共同刻画的单群都满足一些特殊的性质仍是一个开放的问题。[Purposes]To Characterize the alternating group A5,A6 And projective special Linear group L2(7) with the order of group Gand k(G),the number of conjugacy Classes of G.[Methods]Through the relationship between conjugacy length and the order of centralizer of element of finite groups,and the relationship between the prime graph and the structure of finite groups,by classified discussion and the judgment of isomorphism of finite groups.[Findings]Proofed that G≌A5 if |G|=60 and k(G)=5;G≌L2(7) if |G|=168 and k(G)=6;|G|≌A6 if |G|=360 and k(G)=7.[Conclusions]A finite simple group can be characterized by giving the order of Gand k(G),but whether these groups satisfy any property or not is still an open topic.

关 键 词：有限群 共轭类 群阶 共轭类个数 

分 类 号：O152.1[理学—数学]