On ONC-characterization And ON-characterization Of Finite Groups

As is well know,the quantitative characterization of finite groups has always been the focus of group theory research.In the 1980s,Professor Shi Wujie proposed the famous subject of "characterizing finite simple groups with only two orders",and proved that almost all finite simple groups can be fully characterized by "group order"and "set of element order".With the joint efforts of group theorists in China and Russia,this subject was completely solved in 2008.Of course,the key to characterizing the number of groups is the number used With the number of conditions,people naturally expect to reduce the number of conditions as much as possible and use the least number to characterize the most properties of the group.Therefore,how to weaken the quantity conditions and promote the work of Professor Shi Wujie has become an interesting new topic.Since 2009,people used "group order" and "order of higher-order elements" to describe finite groups,as well as "order of highest-order elements","number of highest-order elements" and "set of centralizers of highest-order elements" Three quantities that are only related to the highest-order elements(ie,the first ONC-metric)characterize finite groups and other topics,and have done a series of studies to partially promote the work of Professor Shi Wujie.This article will continue this research,using the first ONC-metric or first ON-metric(the order of the highest order element and the number of the highest order element)characterizes the structure of a finite group,and attempts to promote the work of Professor Shi Wujie and others by changing a certain number of conditions.The full text is divided into four chapters,and the main research results are reflected in Chapters 3 and 4.Chapter 1 mainly introduces the research background,current status,and the main results obtained in this article;Chapter 2 introduces some symbols and lemmas used in this article;The third chapter studies the influence of the first ONC-metric on the structure of the symmetric group Sn(n<13);the fourth chapter discusses the first ON-metric on the single K3-groups and the Li-type single groups L2(q)(where q=pn ?32,p is a prime number)structure.