On Automorphisms and Focal Subgroups of Blocks

Abstract

Given a p-block B of a finite group with defect group P and fusion system on P, we show that the rank of the group is invariant under stable equivalences of Morita type. The main ingredients are the construction, due to Broué and Puig, a theorem of Weiss on linear source modules, arguments of Hertweck and Kimmerle applying Weiss’ theorem to blocks, and connections with integrable derivations in the Hochschild cohomology of block algebras.