Crossed products and blocks with normal defect groups

Blocks having a normal defect group D are fairly well understood. R. Brauer Later Dade [4] extended Brauer's work to study the behavior of arbitrary block idempotents with respect to normal subgroups. Here we use Dade's methods to generalize Reynolds' theorem. Since our reduction produces a crossed product it seems natural to start with crossed products, too. So we work in this slightly more general context. As a corollary we get the following version of Reynolds' result: A. THEOREM. Let G be a finite group and R a complete discrete valuation ring with residue class field F of prime characteristic p. Let B E be a block of the group algebra RG having a normal defect group D, and choose a block b e of RC G (D) with Ee = 0. Denote by G(b) the set of all elements g G such that e g = e, and assume that F is a splitting field for Zb. Then DC G (D)/C G (D) is a normal Sylow p-subgroup of G(b)/C G (D), and B is isomorphic as an R-algebra to S R R DH for some central separable R-algebra S and some twisted group algebra R DH of DH over R where H denotes a complement of DC G (D)/C G (D) in G(b)/DC G (D) and DH the semidirect product of D with H.