On Sum Of The Irreducible Characters Degrees And The ONC-characterizations Of Some Simple Groups

In this thesis,we do research on influence by the sum of the irreducible character degrees of a finite group on its structure,and ONC-characterizations of A14 and A15-1.The influence by the sum of the irreducible character degrees of a finite group on its structure:Given a finite group G,set ? denote the sum of all irreducible characters of G and T(G)=T(1).choose a nontrivial subgroup H of G.let a(?)=[TH,?].For arbitrary ??Irr(H),set T(G)=T(1)=???Irr(H)a(?)?(1),and?(G,H)=T(G)?T(H)=???Irr(H)(a(?)-1)?(1).Let a=a(l?).It follows from H<G that a>1,and hence ?(G,H)>0.In this paper,we discuss the influence on structure of G by the following number 60(G,?)?where?(G?H)=?(G?H)-(a-l)=???Irr(H)(a(?)—1)?(1).?0(G,H)decides the structure of a group to some extent.This research was first done by Berkovich and Mann,they studied the case while ?0(G,?):<2,and got the results that G=(L,H),while?0(G,H)=0,and G is solvable while?0(G,H)<2.The study was published in the Journal of Algebra and attracted the attention of many mathematicians.After that,Yanxiong Yan and professor Guiyun Chen discussed the case while ?0(G,H)=3.As the results have not yet been published,I shall not describe the results here.In this thesis,we continue this work to study the case while ?0(G,H)=4 in chapter 3.2.The ONC-characterizations of A14 and A15.let G be a finite group,o1(G)denote the largest element order of G,n1(G)denote the number of the elements of order o1(G).Assume that G totally has r elements of order 01(G),of which the centralizers are of different orders,and ci(G)(i=1,2,…,r)denote the orders of centralizers of elements of order o1(G).Define the 1st ONC—degree of G as the following sequence of numbers:ONC1(G)={o1(G);n1(G);C1(G),C2 G,…,c,(G)},denoted as ONC1(G).It is proved that K3 simple groups,L2(q)(q=8,11,13,17,19,23,29),Mathieu simple groups and alternating groups An(5?n?13)can be characterized by the 1st ONC—degree,but unfortunately L2(q)(q=16,25)cannot be characterized by the 1st ONC—degree.Since the ONC-degree of alternating groups usually has only 3 numbers,so it is interesting to observe if they can be characterized by the 1st ONC—degree.In chapter 4,we shall prove that A14 can be characterized by the 1st ONC—degree,but we can not prove A15 does.We shall prove if G has a non-connected prime graph and ONC1(G)=ONC1(A15)),then G=A15.