Research On The Influence Of Prime Graph And Average Codimension On The Structure Of Finite Group

In this thesis,we study the relationship between quantities of finite groups and the structure of finite groups,and we investigate the effect of the average codegrees of the irreducible characters on the structure of finite groups,and the relationship between the average codegrees of the irreducible characters and the solvability,super-solvability and nilpotent of finite groups is discussed;Secondly we study the uniqueness with disconnected prime graph for finite simple groups,and we discuss whether the sporadic groups,K3-simple groups and alternating groups can be determined uniquely by their order when they have disconnected prime graph.This thesis is divided into six chapters.Chapter 1 The research background and main results are stated in this thesis.Chapter 2 We introduce some important lemmas needed in this thesis.Chapter 3 We define the average codegrees of the irreducible characters of G as (?) We study the following question:Question Can we find the largest positive integer m,such that all finite groups satisfying acod(G)<m are solvable,super-solvable,and nilpotent,respectively.That is,can we determine the upper bound of m in the three cases?We answer the question and give a perfect upper bound of m which makes the finite group G solvable,super-solvable and nilpotent,respectively.That is,the results with the average codegrees of irreducible characters are parallel to those described by group theorists such as Isaacs by the average degree of irreducible characters.We get the following results:(1)Let G be a finite group.If G is non-solvable,then acod(G)≥ 68/5,and with the equality if and only if G(?)A5.(2)Let G be a finite group.If G is non-supersolvable,then acod(G)≥ 11/4,and with the equality if and only if G(?)A4.(3)Let G be a finite group.If |G| is odd with acod(G)<21/5,then G is nilpotent.Furthermore,we give a general result,(4)let p=min{π(G)}.Then acod(G)<p if and only if G is an elementary abelian p-group.Chapters 4 and 5 We study the following questions:Question.Let G and S be two finite groups with non-connected prime graphs.If |G|=|S| and S is a simple group,is it possible to conclude that G(?)S?The source,theory value and application value of this question lie in:In many quantitative characterizations of simple groups,for a simple group with disconnected prime graph,the proof are usually ascribed to comparing a finite group having disconnected prime graph with a simple group whose prime graph is not connected with the same order,even in some cases,it can be known directly from the assumptions that such two groups are compared.Therefore,if the above question holds for some simple groups with disconnected prime graphs,then the corresponding quantitative characterizations according to this result does not need to prove.In other words,the simple groups hold for the above question can be uniformly generalized to other quantitative characterizations(see follow-up corollaries).The conclusions of these two chapters are as follows:(1)Let G be a finite group,S one of a 26 sporadic groups.If |G|=|S| and the prime graph of G is disconnected,then G(?)S.(2)Let G be a finite group,S an alternating group with non-connected prime graph.If |G|=|S| and the prime graph of G is disconnected,then G(?)S.Combine(1)and(2),we obtain the following corollaries.Corollary 1 Let G be a finite group,S a sporadic group or an alternating group with non-connected prime graph.Then G(?)S if and only if |G|=|S| andπe(G)=πe(S).Corollary 2 Let G be a finite group,S a sporadic group or an alternating group with non-connected prime graph.Then G(?)S if and only if Z(G)=1 and N(G)=N(S).Corollary 3 Let G be a finite group,S a sporadic group or an alternating group with non-connected prime graph.Then G(?)S if and only if G and S have the same order components.Corollary 4 Let G be a finite group,S a sporadic group or an alternating group with non-connected prime graph.Then G(?)S if and only if |G|=|S| and the sets of orders of maximal abelian subgroups of G and S are equal.For K3-simple groups,we can not get the characterization similar to sporadic groups and alternating groups,but we give a complete classification of this group.The conclusions are as follows.(3)Let G be a finite group,S an arbitrary K3-simple group.If |G|=|S|,then the following statements hold:a.If S(?)L2(7)and S(?)U4(2),then G(?)S.b.If S(?)L2(7),the G is isomorphic to one of the following groups:L2(7),C2(?)(C3(?)(C2 × C2 × C7))and C3(?)(C7(?)(C2 × C2 × C2)).c.If S=U4(2),then G(?)U4(2)or G(?)C4(?)(C5(?)D),whereD=(C2 × C2 × C2 × C2)×(C3 × C3 × C3 × C3).Chapter 6 In this chapter,we aim to characterize finite simple groups by using the two conditions of "the average codegrees of the irreducible characters" and“disconnected prime graph”.Since we drop the order equality,this problem becomes difficult.We prove the following theorem.Let G be a finite group with non-connected prime graph,acod(G)=68/5,k(G)=5.Then G(?)A5.