Simple fusion system

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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 p-group is a p-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.