Inductive Sources For π-partial Characters

In this thesis,the inductive sources in the theory of π-partial characters is discussed,and some properties of π-inductive sources are given.After introducing the notion of Bπ-inductive sources and Dπ-inductive sources,a characterization of Iπ-inductive sources is obtained,and the relationship among these three inductive sources is described.The results strengthen Lewis’theorem,cover some of the classical results of inductive sources for complex characters,and also contain the corresponding theorems of inductive sources for Brauer characters.The main results of this thesis are as followsTheorem A Suppose that φ∈Iπ(S),where S≤G andG is π-separable,and let T=Gφbe the stabilizer subgroup of φ in G.If Iπ-character pair(S,φ)is an Iπ-inductive source in G,the following hold.(1)The induction ξ(?)ξG is a bijection Iπ(T|φ)→Iπ(G|φ).(2)Iπ(T|φ)=Iπ(φT),and Iπ(G|φ)=Iπ(φG).We get a characterization of Iπ-inductive sourceTheorem B Suppose that φ∈Iπ(S),where S≤G and G is π-separable.Let φ∈Irr(S)be a standard lift of φ.Then(S,φ)is an Iπ-inductive source in G if and only if the following hold:(1)If 2∈π,then(S,φ)is a Bπ-inductive source in G.(2)If 2(?)π,then(S,φ)is a Dπ-inductive source in G.In general,inductive sources in complex characters have the following basic relations with the Bπ-inductive source and Dπ-inductive source mentioned above.Theorem C Suppose that θ∈Irr(H),where H≤G and G is π-separable,and let(H,θ)be an inductive source in G.Let T=Gθ be the stabilizer subgroup of(H,θ)in G.Then the following hold.(1)If 2∈π,and H(?)G,then(H,θ)is a Bπ-inductive source in G.(2)If 2(?)π,then(H,δ(G,H)θ)is a Dπ-inductive source in G.