2-Sylow子群的阶及元素最高阶和次高阶与A8相同的有限群

On Order of 2-Sylow Subgroup and the Same Largest and Second Largest Element Orders with A8

讨论了群的2-Sylow子群的阶及元素的最高阶和次高阶与A8相同的有限群.利用群的素图及外自同构群、幂零群的若干性质,得出结论:若群G的2-Sylow子群的阶及元素的最高阶和次高阶与A8相同,则存在H■G,使得下列结论之一成立:(i)G/H?A8,|G|=26·3α·5β·7,其中H为3α-2阶且方指数整除3的幂零群,或5β-1阶初等Abel群,或3α-2·5β-1阶且方指数整除15的幂零群;(ii)G/H?L3(4),|G|=26·3α·5β·7,其中H为3β-2阶且方指数整除3的幂零群,或5β-1阶初等Abel群,或3α-2·5β-1阶且方指数整除15的幂零群.由此得到推论:若|G|=|A8|,Ki(G)=Ki(A8)(i=1,2),则G?A8.

In this paper,the structure of finite groups with the same order of 2-Sylow subgroup and the same largest and second largest element orders with A8 has been discussed.The some properties of prime graph components of groups,outer automorphism groups and nilpotent groups are used in discussion.It is concluded that,if the finite groups G has the same order of 2-Sylow subgroup and the same largest and second largest element orders with A8,then a group H■G makes that,(i)G/H?A8,|G|=26·3α·5β·7,and H is a nilpotent group and|H|=3α-2 square index divides 3;or H is an elementary Abel group,and|H|=5β-1;or H is a nilpotent group and|H|=3α-2·5β-1 square index divides 15;(ii)G/H?L3(4),|G|=26·3α·5β·7,and H is a nilpotent group and|H|=3β-2 square index divides 3;or H is an elementary Abel group,and|H|=5β-1;or H is a nilpotent group and|H|=3α-2·5β-1 square index divides 15.As a corollary,it follows that if|G|=|A8|,Ki(G)=Ki(A8)(i=1,2),then G?A8.