Research On Character Degrees Of Finite Groups And Related Topics

The characters of finite groups have a large influence on the structure,for instance,Ito's thereom shows that if the prime p does not divide ?(1)for all ??Irr(G),then G has a normal and abelian Sylow p-subgroup.Related to the character degrees,a well-known conjecture has been addressed by Huppert:If G and M have the same set of character degrees where M is a non-abelian simple group,then G?A× M where A is abelian.Huppert's conjecture has been proved by some scholars true for some simple groups.Also the influence of character degrees on the group structure has been studied by some authors and some good results have been gotten.There is a close relation between these graphs(except the second chapter,but it's related to character).The detail is as follows.Some basic knowledge about finite group theory and the main results are given first.The order of vertices of vanishing prime graph is used to study the structure in Chapter 2.Simple K3-groups by this new concept first posed are characterizable and this method succeeds at least part.In Chapter 3,the graph of character degree power are used to characterize Janko groups.This concept is put forward by Qin et.al because some simple groups are not uniquely de-termined by the character degree graph and it order.Janko groups and certain groups of the form An×An are characterized using this method.In Chapter 4,the influence of the orders of vertices of degree power graph on the struc-ture is given.This concept is first addressed and used to characterize Mathieu-groups and some projective special linear groups.Our method is useful in practice to characterize theses simple groups.In Chapter 5,the maximal subgroup with the character degrees of maximal subgroup is combined first to investigate the structure.The structure of simple groups whose all sub-groups have only irreducible characters of prime-power-degree is determined.