Saturated fusion system: Difference between revisions

Latest revision as of 01:49, 30 April 2012

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Definition

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

Condition name	Qualitative description of condition	Condition details
contains inner fusion system	This condition makes sure that the obvious homomorphisms are included	For any subgroups $Q, R \leq P$ , all injective homomorphisms from $Q$ to $R$ that arise as restrictions of inner automorphisms of $P$ , are present in $F$ . In other words, the inner fusion system on $P$ is a subcategory of $F$ .
Sylow axiom	This condition is a purely global condition, i.e., it gives information only about automorphisms at the global level	The inner automorphisms of $P$ form a Sylow subgroup of the group of automorphisms of $P$ in $F$ .
Extension axiom	This condition links up the local and the global	Call a subgroup $R$ of $P$ fully normalized by $F$ if $\| N_{P} (Q) \| \leq \| N_{P} (R) \|$ for any $Q ≅ R$ where $≅$ is isomorphism in the category $F$ . Also define, for any morphism $φ : Q \to P$ in $F$ : $N_{φ} = {y \in N_{P} (Q) ∣ \exists z \in N_{P} (φ (Q)), φ (y u y^{- 1}) = z φ (u) z^{- 1} \forall u \in Q}$ Then the statement of the extension axiom is: Every morphism $φ : Q \to P$ such that $φ (Q)$ is fully $F$ -normalized, extends to a morphism $ψ : N_{φ} \to P$ .

Terminological note

Some people use the term fusion system for what is defined here as a saturated fusion system. Others use fusion system to mean a category on a finite p-group that contains the inner fusion system. People in both terminological camps are usually interested only in saturated fusion systems when they talk of fusion systems.

There are multiple conventions on whether the term fusion system should refer to saturated fusion system or to a weaker notion. For our purposes, we will mostly be interested in saturated fusion systems, because these mimic/generalize situations where the
$p$
-group is a
$p$
-Sylow subgroup of some finite group. Unless otherwise specified, we are referring to saturated fusion systems when we talk about fusion systems.

References

Introduction to Fusion Systems by Markus Linckelmann^Weblink^{More info}

@@ Line 3: / Line 3: @@
 ==Definition==
-Let <math>P</math> be a [[group of prime power order]], say a finite <math>p</math>-group, for a prime <math>p</math>. A '''fusion system''' <math>\mathcal{F}</math> on <math>P</math> is a [[category on a finite p-group|category]] on <math>P</math> with the following properties:
+Let <math>P</math> be a [[group of prime power order]], say a finite <math>p</math>-group, for a prime <math>p</math>. A '''saturated fusion system''' <math>\mathcal{F}</math> on <math>P</math> is a [[defining ingredient::category on a finite p-group|category]] on <math>P</math> (in the sense of a [[category on a finite p-group]]) with the following properties:
-* For any subgroups <math>Q, R \le P</math>, all injective homomorphisms from <math>Q</math> to <math>R</math> that arise as restrictions of [[inner automorphism]]s of <math>P</math>, are present in <math>\mathcal{F}</math>.
+{| class="sortable" border="1"
-* The inner automorphisms of <math>P</math> form a Sylow subgroup of the group of automorphisms of <math>P</math> in <math>\mathcal{F}</math>.
+! Condition name !! Qualitative description of condition !! Condition details
-* It satisfies the extension axiom. The extension axiom is as follows:
+|-
+| contains [[inner fusion system]] || This condition makes sure that the ''obvious'' homomorphisms are included  || For any subgroups <math>Q, R \le P</math>, all injective homomorphisms from <math>Q</math> to <math>R</math> that arise as restrictions of [[inner automorphism]]s of <math>P</math>, are present in <math>\mathcal{F}</math>. In other words, the [[inner fusion system]] on <math>P</math> is a subcategory of <math>\mathcal{F}</math>.
+|-
+| Sylow axiom || This condition is a purely ''global'' condition, i.e., it gives information only about automorphisms at the global level || The inner automorphisms of <math>P</math> form a Sylow subgroup of the group of automorphisms of <math>P</math> in <math>\mathcal{F}</math>.
+|-
+| Extension axiom || This condition links up the local and the global || Call a subgroup <math>R</math> of <math>P</math> [[defining ingredient::subgroup fully normalized by a category|fully normalized]] by <math>\mathcal{F}</math> if <math>|N_P(Q)| \le |N_P(R)|</math> for any <math>Q \cong R</math> where <math>\cong</math> is isomorphism in the category <math>\mathcal{F}</math>.<br>Also define, for any morphism <math>\varphi:Q \to P</math> in <math>\mathcal{F}</math>:<br><math>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 \}</math><br>Then the statement of the ''extension axiom'' is:<br>Every morphism <math>\varphi:Q \to P</math> such that <math>\varphi(Q)</math> is fully <math>\mathcal{F}</math>-normalized, extends to a morphism <math>\psi:N_\varphi \to P</math>.
+|}
-Call a subgroup <math>R</math> of <math>P</math> [[subgroup fully normalized by a category|fully normalized]] by <math>\mathcal{F}</math> if <math>|N_P(R)| \le |N_P(Q)|</math> for any <math>Q \cong R</math> where <math>\cong</math> is isomorphism in the category <math>\mathcal{F}</math>.
+==Terminological note==
-Also define, for any morphism <math>\varphi:Q \to P</math> in <math>\mathcal{F}</math>:
+Some people use the term ''fusion system'' for what is defined here as a saturated fusion system. Others use [[fusion system]] to mean a [[category on a finite p-group]] that contains the inner fusion system. People in both terminological camps are usually interested only in saturated fusion systems when they talk of fusion systems.
-<math>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 \}</math>
+{{saturated fusion system note}}
-Then the statement of the ''extension axiom'' is:
-Every morphism <math>\varphi:Q \to P</math> such that <math>\varphi(Q)</math> is fully <math>\mathcal{F}</math>-normalized, extends to a morphism <math>\psi:N_\varphi \to P</math>.
 ==References==
 * {{paperlink|Fusion-intro}}