The Group Properties Wiki (pre-alpha)

TIP: Get tips on handling doubts and solving riders

ABOUT US: We use a Creative Commons license. All our content is free to reuse, with attribution. Learn more

ALSO CHECK OUT: Commalg: The Commutative Algebra Wiki

Fusion system

From Groupprops

Jump to: navigation, search
This is a variation of group
View a complete list of variations of group OR read a survey article on varying group

Definition

Let P be a group of prime power order, say a finite p-group, for a prime p. A fusion system \mathcal{F} on P is a category on P with the following properties:

  • For any subgroups Q, R \le P, all injective homomorphisms from Q to R that arise as restrictions of inner automorphisms of P, are present in \mathcal{F}
  • It satisfies the extension axiom. The extension axiom is as follows:

Call a subgroup R of P fully normalized by \mathcal{F} if |N_P(R)| \le |N_P(Q)| for any Q \cong R where \cong is isomorphism in the category \mathcal{F}.

Also define, for any morphism \varphi:Q \to P in \mathcal{F}:

N_\varphi = \{ y \in N_P(Q) \mid \exists z \in N_p(\varphi(Q)), \varphi(yuy^{-1} ) z\varphi(u)z^{-1} \ \forall u \in Q \}

Then the statement of the extension axiom is:

Every morphism \varphi:Q \to P such that \varphi(Q) is fully \mathcal{F}-normalized, extends to a morphism \psi:N_\varphi \to P.

References

Personal tools