Primitive Inducors Of Character Triples

In this paper,we study the character triples as objectives from the point of view of the theory of category.We define the normality and subnormality of subtriples to replace the nilpotency condition on groups.We prove that any two primitive inductors of the given character triple have the same degree.This generalizes an inductor theorem of Isaacs.As an application,we give a condition such that two primitive inductors of a irreducible complex character have the same degree,and this also strengthens the corresponding results of Isaacs.The main conclusions of this thesis are as follows:Theorem A Suppose that T =(G,N,?)is a character triple,where N is solvable.Suppose further that :(1)Every primitive sub-triple of T is subnormal.(2)All maximal normal restrictors of every primitive sub-triple of T are primitive.Then any two primitive inductors of T have the same degree.As an application of Theorem A,we can obtain Theorem B of this paper by simplifying the condition(2)of theorem A.Theorem B Suppose that T =(G,N,?)is a character triple,where N is solvable.If every primitive sub-triple of T is nilpotent and subnormal,then any two primitive inductors of T have the same degree.As an application of Theorem B in this paper,the result of the following Theorem C also generalizes Theorem B of Isaccs in [9].Theorem C Suppose that G is an arbitrary group and ? ? Irr(G).Suppose that G has a solvable normal subgroup N such that ? = ?Nis irreducible,and every primitive subtriple of character triple T =(G,N,?)is nilpotent and subnormal.Then all primitive characters which induce ? have the same degree.