Inductive Sources And Compound Clifford Correspondence Of Characters

The discrimination problem for inductive sources was studied and some necessary and sufficient conditions for a character pair to be an inductive source were obtained,where an inductive source of G is a character pair(H,?),such that Ind : Irr(G_?|?)?Irr(G|?)is a bijection.A theorem of Dade regarding inductive sources and compound Clifford correspondence was strengthened and generalized under some oddness conditions.In particular,this not only simplifies Dade's original proof and is independent on the theory of hyperbolic modules,but also is a purely character theoretic proof.The main conclusions of this paper are as follows:Theorem A Let G be a finite group and suppose that(A,?)? IS(G)is an inductive sources of G.Let(A,?)?(H,?)? G.Then(H,?)is also an inductive sources of G if and only if one of the following equivalent conditions holds:(1)(H_?,?_?)? IS(G_?)and G_?(?_?)? N_G(H).(2)(H_?,?_?)? IS(G_?)and G(?,?)= G(?_?).(3)(H_?,?_?)? IS(G)and G(?,?)= G(?_?).(4)Ind : Irr(G(?,?)|?_?)? Irr(G|?_?)is bijection.Theorem B Let T =(G,N,?)be a character triple.Suppose that N is solvable and ?(1)is odd.Then any inductive source correspondent of T is a compound Clifford correspondent,i.e.,ISC(T)= CCC(T).