# Fusion systems and localities -- a dictionary

## TL;DR

This paper establishes a correspondence between partial normal subgroups of localities and normal subsystems of fusion systems, creating a dictionary that links concepts in both frameworks and enabling new proofs and results.

## Contribution

It introduces a one-to-one correspondence between partial normal subgroups and normal subsystems, facilitating translation of concepts and deriving new results in fusion system theory.

## Key findings

- Established a dictionary linking localities and fusion systems concepts
- Provided new proofs for existing theorems in fusion systems
- Introduced a notion of product of normal subsystems in saturated fusion systems

## Abstract

Linking systems were introduced to provide algebraic models for $p$-completed classifying spaces of fusion systems. Every linking system over a saturated fusion system $\mathcal{F}$ corresponds to a group-like structure called a locality. Given such a locality $\mathcal{L}$, we prove that there is a one-to-one correspondence between the partial normal subgroups of $\mathcal{L}$ and the normal subsystems of the fusion system $\mathcal{F}$. This is then used to obtain a kind of dictionary, which makes it possible to translate between various concepts in localities and corresponding concepts in fusion systems. As a byproduct, we obtain new proofs of many known theorems about fusion systems and also some new results. For example, we show in this paper that, in any saturated fusion system, there is a sensible notion of a product of normal subsystems.