The Character Correspondences on pi-Separable Groups

In this paper, we mainly consider the relationship between the complex irreducible characters of a π-separable group and the complex irreducible characters of its Hall π-subgroup. If a π-group S acts on a π-separable group G, let H be an S-invariant Hall π-subgroup of G and CNG(H)/H(S)=1. Then we construct a natural bijection from the set LinS(H) onto the set Irrπ′,S(G). Furthermore, we get a bijection from the linear characters of H onto Irrπ′(G).