Fusion system

Definition
The term fusion system is used for some additional structure (a category structure) on a finite p-group satisfying some conditions. There are two conflicting definitions of fusion system in use.

Definition followed by Broto-Levi-Oliver
Broto-Levi-Oliver, Kessar, and others use the following convention: they define a fusion system as a category on a finite p-group that contains the inner fusion system. However, they do not include the two other axioms -- the Sylow condition and the extension axiom -- that form part of the saturated fusion system definition.

The two definitions are presented and contrasted below for $$P$$ a group of prime power order, say a finite $$p$$-group, for a prime $$p$$, and $$\mathcal{F}$$ a category on $$P$$:

Definition followed by Linckelman
Fusion systems as defined by Linckelman are what we here call saturated fusion systems. These are fusion systems that also satisfy the Sylow axiom and the extension axiom.

Fusion systems induced by groups
Given any finite group $$G$$ and any $$p$$-subgroup $$P$$ of $$G$$, we can define the fusion system induced by $$G$$ on $$P$$. A case of special interest is where $$P$$ is a $$p$$-Sylow subgroup of $$G$$, and in that case we talk of the fusion system induced by a finite group on its p-Sylow subgroup. The following are true for fusion systems induced by finite groups on $$p$$-subgroups: