Character correspondences in p-solvable groups (Q2640694)

Let p be a prime, A be a finite p-group acting on a p-solvable group H, B be a p-block of H of full defect. Denote the set of A-invariant irreducible complex characters of height 0 of B by \(X_ 0(A,B)\). Suppose that \(X_ 0(A,B)\neq \emptyset\). The main result is following: Theorem. There exists a finite group K and a p-block b such that: (i) \(| K| \leq | H|\); (ii) a Sylow p-subgroup of K is Abelian and normal in K and is a homomorphic image of a Sylow p-subgroup of H; (iii) \(| X_ 0(A,B)| =k(b)\), where k(b) is the number of irreducible complex characters of b. This theorem is a generalization of the well- known Glauberman correspondence. It may be used in inductive arguments about the set \(X_ 0(A,B)\).