Study On The Quantitative Properties And Group Structure Of Some Finite Non-abelian Simple Groups

This paper studies the quantitative characterization of two types of problems:one is to study the relationship between the degrees of irreducible character and the structure of group.The second is to study the ON C-Characterization of the alternating group A₂₁.1.The influence by irreducible character degrees of a finite group on the struc-ture of the finite group.Using the arithmetic properties of the irreducible characteristic degrees to char-acterize the structure of a finite groups is a classic topic in finite group representation theory.For example,for any??Irr?G?,the set of all irreducible characters of group G,we have??1?|G|.However,it is an interesting and difficult problem that how these arithmetic properties to affect the structure of the group.There were so many famous results appeared in such field.For example,Ito-Michler pointed out that if for any??Irr?G?,there exist a prime p such that p??1?,then G has an abelian sylow p-subgroup.Aslo,G.Y.Chen in 1994 proved that any non-abelian simple group can be uniquely determined by its characteristic table.The first type of problem in this article continues to study the quantitative characterization of single groups.This problem is related to the following Huppert conjecture:Huppert's conjecture:Let H be a non-abelian simple group and G be a group such that cd?G?=cd?H?,then G?=H×A,where A is an abelian group.So far,Huppert's conjecture is still open.Huppert's conjecture needs to con-sider all the irreducible characteristic degrees of the group,and so this condition is very strong.In 2011,Chen Guiyun and Xu Haijing weakened the condition of Huppert's conjecture,and for the first time they put forward the following topic,that is,using the order of a finite group and some large irreducible character degrees of the group to characterize non-abelian simple groups,and successfully character-ize simple K₃-groups,some sporadic simple groups and some simple group of Lie type.We continue to their research on this problem and prove that the simple groups L₂?43?,L₂?61?,L₂?67?,L₂?83?,L₂?101?,L₂?103?,L₂?113?,L₂?121?,L₂?125?,L₂?41?,L₂?59?,L₂?71?,L₂?79?,L₂?89?,L₂?109?,U₃?13?,U₃?16?,U₃?25?,U₃?32?,L₃?13?,U₃?17?,L₃?19?,U₆?2?,U₃?9?and U₃?23?can be uniquely determined by their orders and one largest irreducible character degree,and the simple groups O₅?4?,O₅?9?,L₃?27?,U₄?4?,A₁₁,G₂?5?,M^cL,L₃?31?and L₄?5?can be uniquely determined by their orders and two largest irreducible character degree,and the simple groups L₂?64?,G₂?4?and L₃?9?can be uniquely determined by their orders and at most three largest irreducible character degrees.2.The ON C-characterization of the alternating group A₂₁Wujie Shi put forward the problem of”two-order characterization”,that is,”any non-abelian finite simple group can be uniquely determined by their order and the set of all the orders of the elements of the group”.After then,quantitative characterization of a finite group became a hot topic in group theory.Shi's problem has been successfully solved by many mathematicians.Inspired by this problem,Chen Guiyun and He Liguan considered the first ON C-degree problem.Let G be a finite group,o₁?G?denotes the order of the largest order of elements in G,and n₁?G?denotes the number of largest order of elements.Assume that G has exactly r elements of largest order,whose centralizers have distinct orders,say,they are c_i?G?,i=1,2,···,r.LetON C?G?={o₁?G?;n₁?G?;c₁?G?,c₂?G?,···,c_r?G?},We called ON G?G?the 1st ON C-degree.He Liguan proved that many finite non-abelian simple groups can be characterized by the 1st ON C-degree,and especially he proved that any alternate simple group of rank not more than 13 can be characterized by the 1st ON C-degree.After then,Wang Zhongbi continued to his research and obtained that the alternating group A₁₄can be characterized by the1st ON C-degree.Furthermore,if the prime graph of a finite group is unconnected graph,then the alternating simple groups A_n?15 n 20?can be characterized by the 1st ON C-degree.In this paper,we continues to this investigation and show that the alternating group A₂₁can be characterized by the 1st ON C-degree.