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Fusion system
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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 LinckelmannWeblinkMore info