A New Characterization Of The Automorphism Groups Of Sporadic Simple Groups And Suzuki-Ree Groups

In group theory, to study the structure of groups, we always hope to describe a group by using the basic characters of the group. The characterization of quantity was put forward by Professor wujie Shi in1980who did a lot of research on this topic. Afterwards, various quantitative characterizations of groups had been put forward, the topic was intended to describe groups by quantitative conditions on the groups.1996, Guiyun Chen defined the concept order components of a finite group and got the order components of non-abelian simple groups with non-connected prime graphes. He proved that all sporadic simple groups can be uniquely determined by their order components. Except for sporadic simple groups, some finite simple groups were characterized by their order components. Because finite group has no more than six components, the prime graph of any can use not more than seven quantities to characterize a group. Moving by Chen’s work, many foreigners are interested in this topic, and they have done much works on this topic and published about40articles up to now. This paper concerns about what kind of groups can be determined by their order components except for finite simple groups.In the forth part of this paper, we prove that all automorphism groups(except for J2, Mcl) of sporadic simple groups can be characterized by their order components, and obtain the following theorem:Theorem4.1Let G is a group, M be sporadic simple groups (except for J2, Mcl), then G≌Aut(M) if and only if OC(G)=OC(Aut(M)). In the fifth part of this paper, we prove that the automorphism groups of Suzuki-Ree groups can be characterized by their order components, and we have:Theorem5.1Let G be a group, M be the Suzuki-Ree groups2F4(q)(q=2f),2G2(q)(q=3f) and the Tits simple group2F4(2)’. where f=3s, s is a positive integer, then G≌Aut(M) if and only if OC(G)=OC(Aut(M)).