Tate's and Yoshida's theorem on control of transfer for fusion systems

TL;DR

This paper extends classical theorems of Tate and Yoshida to fusion systems, introducing new concepts like $p$-group residuals and cohomological transfer maps, leading to a $p$-nilpotency criterion.

Contribution

It develops analogues of Tate and Yoshida's control of transfer results within the framework of fusion systems, incorporating novel notions such as $p$-group residuals and transfer maps in cohomology.

Findings

Established analogues of Tate and Yoshida's theorems for fusion systems.

Introduced $p$-group residuals and transfer maps in cohomology for fusion systems.

Derived a $p$-nilpotency criterion from the new theoretical framework.

Abstract

We prove analogues of results of Tate and Yoshida on control of transfer for fusion systems. This requires the notions of $p$-group residuals and transfer maps in cohomology for fusion systems. As a corollary we obtain a $p$-nilpotency criterion due to Tate.