Fusion system induced by a finite group on its p-Sylow subgroup is a saturated fusion system
Suppose is a finite group, is a prime number, and is a -Sylow subgroup of . Suppose is defined as a category with objects the subgroups of and morphisms given as follows: For every and subgroups such that , there is a morphism given by . This is a category on and is in fact a fusion system.
Proof that it is a category
First, we prove that we get a category on .
- It is closed under composition: Suppose and are morphisms induced by conjugation by respectively. Then, and , so . Thus, there is a morphism in from to given by conjugation by . We see that this is the composite .
- All morphisms are group homomorphisms: This follows because each morphism arises as a restriction of an inner automorphism on the whole group.
- It contains all inclusion maps: This follows because we can always set to be the identity element.
- If is a morphism, then the restriction , and its inverse, are morphisms. This follows from the definition.
Proof that it satisfies the conditions for a fusion system
We verify the three conditions:
- All morphisms induced by inner automorphisms from are present: This is true, because in fact all morphisms induced by inner automorphisms from are present.
- The inner automorphisms of form a -Sylow subgroup of : comprises the automorphisms induced by elements of . Since is -Sylow in , it is -Sylow in , so since Sylow satisfies image condition, the image is -Sylow in .
- The extension axiom: PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]