Lewis Correspondence And Some Properties Of Magic Character

Let (G,K,L,ε,φ) be a basic configuration with stabilizing complement H. In the character theory of finite groups , a foudamental and important problem is that whether there exsit a one-to-one correspondence: Δ_C~G : Irr(H|φ) →Irr(G|ε).Let (G,K,L,ε,φ) be a coprime fully-ramified basic configuration ,Lewis proved it had a stabilizing complement in [2] , further defined a central H-complete chain C, used it to construct a bijection Δ_C~G : Char(H|φ → Char(G|ε), which is called Lewis correspondence in this paper;using this central H-complete chain C, Lewis defined a magic character Ψ_C ∈ Char(H) such that Δ_C~G can be uniquely determined by Ψ_C, that is if θ ∈ Irr{H|φ) and x ∈ Irr(G|ε), then x = Δ_C~G (θ) if and only if xH =Ψ_Cθ. In order to convenient ,we call this magic character Lewis magic character in this paper.It is mainly discussed that the performance of Lewis correspondence Δ_C~G and Lewis magic character Ψ_C under restrictions of character, proved respectively Δ_C~G respect restriction and Ψ_C has inheritance, that is Lewis correspondence Δ_C~G commutes with restrictions of character and restrictions to subgroups of Lewis magic character Ψ_C are still magic. Furthermore , we give a property of Lewis chain in this paper,obtain a inequality about calculation problem of the character, then strengthen some related results of Lewis.The following are main results of this thesis, the first result is Lewis correspondence respect restriction:Theorem 2.1: Let (G,K,L,ε,φ) be a coprime fully-ramified basic configuration with stabilizing complement H. Assume that C is a central H-complete chain in K starting with (L,φ).Let U is a subgroup of G,L ≤ U ≤ H,V = KU, then Δ_C~G: Char(H|φ)→ Char(G|ε) and Δ_C~V : Char(U|φ) → Char(V|ε) are bijections. Futhermore, If θ ∈ Irr{H|φ), then ((Δ_C~G(θ))_V = Δ_C~V(θ_U).Secondly,we discuss the inheritance of the Lewis magic character:Theorem 2.2: Let (G, K, L, ε,φ) be a coprime fully-ramified basic configuration with stabilizing complement H. Assume that C is a central H-complete chain in K starting with (L, φ),Ψ_C is the Lewis magic character associated with C.Let U is a subgroup of G,L ≤ U ≤ H,V = KU, then (Ψ_C)_U is the Lewis magic character associated with (V, K, L, ε, φ) and chain C.The last result is a property of Lewis chain ,gives a inequality about calculation of the character:Theorem 2.3: Let chain C: {L,