The Mckay Conjecture For π-partial Characters

In this thesis,we study The McKay conjecture for Isaacs’ π-partial characters,and obtain a canonical bijection between the two sets of corresponding irreducible π-partial characters based on a related theorem of Wolf.This result can be viewed as a π-theoretic version of McKay conjecture for monomial characters in which Isaacs established a canon-ical bijection.The main results of this thesis are as followsTheorem A Let G be a π-separable group and let H∈Hallπ(G).Then there exists a bijectionf:{φ∈Iπ(G)|φ(1)is a π’-number}→{ξ∈Iπ(NG(H))|ξ(1)is a π’-number}such that if ψ∈Iπ(G)and φ(1)is a π’-number,then f(φ)=(λHU(H))NG(H),where H(?)U(?)G,λ∈Iπ(U)is a linear character,such that φ=λG.Furthermore,f(φ)is independent of the choice of U and λ,and every Iπ-character in these two sets is a monomial Iπ-character.Putting π={p}’,we can obtain the McKay conjecture for Brauer characters from Theorem A.Corollary B Let G be a p-solvable group and let Q be a p-complement of G.Then there exists a bijectionf:{φ∈IBr(G)|φ(1)is a p-number}→{ξ ∈IBr(NG(Q))|ξ(1)is a p-number}such that if φ∈IBr(G)and φ(1)is a p-number,then f(φ)=(λNU(Q))NG(Q),where Q(?)U(?)G and λ∈IBr(U)is a linear character with φ=λG.Furthermore,f(φ)is independent of the choice of U and λ,and every Brauer character in these two sets is a monomial Brauer character.Putting π={p}’,we can obtain another dual version of π-theoretic of McKay con-jecture from Theorem A.Corollary C Let G be a p-solvable group and let P∈Sylp(G).Then there exists a bijectionf:{φ∈Ip(G)|φ(1)is a p’-number}→{ξ∈Ip(NG(P)|ξ(1)is a p’-number}suck that if φ∈Ip(G0 and φ(1)is a p’-number,then f(φ)=(λNu(P)NG(P),where P(?)U(?)G,λ∈Ip(U)is a linear character,such that φ=λG.Furthermore,f(φ)is independent of the choice of U and λ,and every Ip-character in these two sets is a monomaial Ip-character.