Glauberman Correspondence of p-Blocks of Finite Groups,

Let G and A be finite groups with coprime orders. Suppose that A is solvable and that it acts on G by automorphisms. Let C = CG(A). By Irr(G) and IrrA(G) we denote the set of all irreducible and all A-invariant irreducible characters of G, respectively. Let D ≤ C be a fixed p-subgroup of G for a prime p. Using the Glauberman correspondence A. Watanabe (J. Algebra216 (1999), 548–565) recently established a Glauberman correspondence between A-invariant p-blocks B of G with defect group D and p-blocks B1 of C with defect group D. Let Br(B) and Br(B1) be the Brauer correspondents of B and B1 in NG(D) and NC(D), respectively. The main result of this article asserts that the block algebras Br(B) and Br(B1) are Morita equivalent. Furthermore, if G is p-solvable and D is abelian, then the block algebras B and B1 are Morita equivalent.