Induction and Character Correspondences in Groups of Odd Order,

Let P be a Sylow p-subgroup of G. By Irrp′(G), we denote the set of irreducible characters of G which have degree not divisible by p. When G is a solvable group of odd order, M. Isaacs constructed a natural one-to-one correspondence *:Irrp′(G) → Irrp′(NG(P)) which depends only on G and P. In this paper, we show that if ξG = χ Irrp′(G), then (ξ*)NG(P) = χ*.