On automorphisms and focal subgroups of blocks

TL;DR

This paper investigates the invariance of the rank of the quotient of a defect group by its focal subgroup under stable Morita equivalences, using advanced algebraic tools and cohomological methods.

Contribution

It establishes the invariance of the rank of P/foc(F) under stable equivalences of Morita type for blocks of finite groups, linking fusion systems and Hochschild cohomology.

Findings

Rank of P/foc(F) is invariant under stable Morita equivalences.

Uses star-construction and Weiss' theorem in the analysis.

Connects focal subgroups with Hochschild cohomology.

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.