Crossed products and blocks with normal defect groups

Abstract

Blocks having a normal defect group D are fairly well understood.R. Brauer [3] reduced the analysis of their block idempotents to blocks of defect zero in the quotient of the centralizer of D by the center of D. A reduction for the ordinary and modular characters in such blocks to the twisted group algebra of a naturally defined group having D as normal Sylow p-subgroup was given by Reynolds [10].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 γ DH for some central separable R-algebra S and some twisted group algebra R γ 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.

Locations

• CiteSeer X (The Pennsylvania State University) - View - PDF
• Communications in Algebra - View