The Character Correspondence Relationships On Finite Groups

In the theory of finite groups,it contains profound and important informationto research the relationship between the global and local structure of a finite groupG.This relationship represents the connection of characters between the group Gand its subgroups in the theory of characters of finite groups,and this is mainlyreflected in the connection of irreducible characters between the group G and itssubgroups,especially the dynamic expressions about the induction,restriction andextention of characters.One of the expressions is the character correspondence.The theory of character correspondeces is an important topic in the theory ofcharacters of finite groups.There are numerous literature and results about it atpresent.In this paper,we firstly research in the properties and correspondencesof characters ofÏ€-separable groups under the group actions,and get the followingresult:Let aÏ€-group S act on aÏ€-separable group G,H be an S-invariant HallÏ€-subgroup of G,C_NG(H)/H(S)=1.Ifλ∈Lins(H),x∈Irr_{π′.S}(G),then there existsa character correspondence:f:Lin_S(H)→Irr_{π′,S}(G)λ→xwhere [λ,xH]≠0.Moreover,there exists a correspondence f′:Irr₍π′,S(N_G(H))→Irr₍π′,S(G),θ→x,whereθH and xH have a common S-invariant linear constituent.Under the condition of normal triple,we also give the following correspondenceof Brauer characters:Let G be a finite group,p-solvable subgroup X (?)G.(X,N,M)be a coprimenormal triple and H be its complement.N/M be a p′-group,modular character pair(M_cψ)(?)(N,(?)).If H satisfies one of the following conditions:(1)C_N/M(H)=N/M;(2)H(?)I_G((?))∩I_G(ψ), then there exists a Brauer character correspondence *:IBr(H|ψ)→IBr(X|(?)),ζ→ζ^*.such that (ξ(1))/(ψ(1))=(ξ^*(1))/((?)(1)).The main contents of this paper are arranged as follows:In the introduction,we provide some researching development related to thetheory of characters and representation of finite groups,and explain the backgroudsand research methods of this paper in detail.In chapter one,we give the basic concepts of representations and characters offinite groups firstly.Moreover,we provide the definition and properties of parallel-ogram configurations,and discuss the character correspondences about this config-urations.In chapter two,we get the properties and correspondence of characters ofÏ€-separable groups under the group actions,and derive several significant corollaries.In chapter three,we discuss the properties of normal triple and the dynamicexpressions of Brauer characters.Furthermore,we get the correspondence relation-ships of Brauer characters.