A Krull-Remak-Schmidt theorem for fusion systems

TL;DR

This paper establishes a unique factorization theorem for saturated fusion systems over discrete p-toral groups, akin to the classical Krull-Remak-Schmidt theorem, with conditions for uniqueness based on the system's center and focal subgroup.

Contribution

It extends the Krull-Remak-Schmidt theorem to fusion systems, providing conditions for the uniqueness of their indecomposable factorization.

Findings

Unique factorization of saturated fusion systems into indecomposables.

Conditions for uniqueness include trivial center or full focal subgroup.

Motivated by automorphism group questions of product fusion systems.

Abstract

We prove that the factorization of a saturated fusion system over a discrete $p$-toral group as a product of indecomposable subsystems is unique up to normal automorphisms of the fusion system and permutations of the factors. In particular, if the fusion system has trivial center, or if its focal subgroup is the entire Sylow group, then this factorization is unique (up to the ordering of the factors). This result was motivated by questions about automorphism groups of products of fusion systems.