Primitiveness Of Character Triples And X-Nilpotent Group

Isaacs found the theorem of character stabilizer limits, in which he introduced the concepts of inductor and restrictor. With the two concepts, he proved two main results. One is the relation between the primitive character triples and quasiprimitive character triples. The other is certainty relations of the inductors between character triples and its restictors.The purpose of this thesis is to generalize the above two results. About the first problem, we prove every primitive character triple is a quasiprimitive character triple. The following is the first main result:Theorem 2.2 Let (G, N, θ) be a primitive character triple. Then it is a quasi-primitive character triple.After introducing the concept of x-nilpotent group, we generalize a result of Isaacs's:Theorem 2.4 Let (G,N,θ) be a quasi-primitive character triple. If N is a θ-nilpotent group, then (G, N, θ) is a primitive character triple.We intrduce a new concept of c-nilpotent group, and discuss the nature of c-nilpotent group. We prove quotient groups also have the nature. This is the third result:Theorem 2.6 Let G be a c-nilpotent group, N (?)G. Then G/N is a c-nilpotent group.About the second problem, we discuss the converse of Isaacs's result, and prove the following result:Theorem 2.12 Let (G, N, θ) be a character triple, and (S, D, γ) be its a restrictor. If (S, D,γ) is a primitive character triple, then (G, N, 6) is a primitive character triple.