On automorphisms and focal subgroups of blocks

Abstract

Given a p-block B of a finite group with defect group P and fusion system F on P we show that the rank of the group P/foc(F) is invariant under stable equivalences of Morita type. The main ingredients are the star-construction, due to Broue 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.