Vertices Of Irreducible Complex Characters

Let G be a finite group,and ??Irr(G)be an irreducible complex character.As an analogue of the vertex of irreducible Brauer characters,how to define the vertex of ? is an important problem in the current group representation theory.In order to unify several different vertex definitions,Cossey gave a generalized vertex construction in 2008 and proved that the vertex is unique under conjugation in odd-order groups.In this thesis,the oddness condition on groups is weakened and the Cossey vertex is also unique under linear conjugation in the more general ?-separable groups.It also includes the classical vertex subgroups when restricting to ?-elements,thus generalizing the main result of Cossey and broadening the application scope of vertex theory.The main results of this thesis are as follows:Theorem A.Let G be ?-separable,where 2(?)?.Assume that ??Irr(G)is a lift of??I?(G)and ?(1)is an odd number.Then the Cossey vertices(Q,?)of ? are linearly conjugate.As an application of Theorem A,we obtain the following Corollary B.Corollary B.Let G be ?-separable,where 2(?)?.Assume that ??Irr(G)is a lift of??I?(G)and ?(1)is an odd number.Then the set of all Cossey vertex subgroups of ? is exactly the set of all vertex subgroups of ?.