# Simple fusion system

From Groupprops

Template:Fusion system property

There are multiple conventions on whether the term fusion system should refer to saturated fusion system or to a weaker notion. For our purposes, we will mostly be interested in saturated fusion systems, because these mimic/generalize situations where the -group is a -Sylow subgroup of some finite group.Unless otherwise specified, we are referring to saturated fusion systems when we talk about fusion systems.

## Definition

A fusion system on a nontrivial group is termed a **simple fusion system** if it has no proper nontrivial normal fusion subsystems. In other words, the only normal fusion subsystems are the whole fusion system and the trivial fusion system on the trivial subgroup.