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Definition

Let P be a group of prime power order, say a finite p-group, for a prime p. A fusion system on P is a category on P (in the sense of a category on a finite p-group) with the following properties:

• For any subgroups , all injective homomorphisms from Q to R that arise as restrictions of inner automorphisms of P, are present in .
• The inner automorphisms of P form a Sylow subgroup of the group of automorphisms of P in . In other words, the inner fusion system is a subcategory of .
• It satisfies the extension axiom. The extension axiom is as follows:

Call a subgroup R of P fully normalized by if for any where is isomorphism in the category .

Also define, for any morphism in :

Then the statement of the extension axiom is:

Every morphism such that is fully -normalized, extends to a morphism .

References

• Introduction to Fusion Systems by Markus Linckelmann^{WeblinkMore info}