Control of fixed points and existence and uniqueness of centric linking systems

TL;DR

This paper provides a classification-free proof of the Martino-Priddy conjecture by removing the reliance on the classification of finite simple groups, using control of fixed points in $p$-local subgroups.

Contribution

It introduces a new approach to prove the Martino-Priddy conjecture without assuming the classification of finite simple groups, extending previous results to all primes.

Findings

Classification-free proof of the Martino-Priddy conjecture.

Application of control of fixed points to $p$-local subgroups.

Extension of results to $p=2$ with offenders.

Abstract

A. Chermak has recently proved that to each saturated fusion system over a finite $p$-group, there is a unique associated centric linking system. B. Oliver extended Chermak's proof by showing that all the higher cohomological obstruction groups relevant to unique existence of centric linking systems vanish. Both proofs indirectly assume the classification of finite simple groups. We show how to remove this assumption, thereby giving a classification-free proof of the Martino-Priddy conjecture concerning the $p$-completed classifying spaces of finite groups. Our main tool is a 1971 result of the first author on control of fixed points by $p$-local subgroups. This result is directly applicable for odd primes, and we show how a slight variation of it allows applications for $p=2$ in the presence of offenders.