Normal fusion subsystem: Difference between revisions

Latest revision as of 03:17, 8 July 2013

ANALOGY: This is an analogue in fusion system of a property encountered in group. Specifically, it is a fusion subsystem property analogous to the subgroup property: normal subgroup
View other analogues of normal subgroup | View other analogues in fusion systems of subgroup properties (OR, View as a tabulated list)

Definition

A fusion subsystem $G$ of a fusion system $F$ on a group of prime power order $P$ is termed a normal fusion subsystem if:

The subgroup $Q$ of $P$ for which $G$ is a fusion system is a strongly closed subgroup of $P$ . In other words, for any $φ : R \to P$ with $φ \in F$ , $φ (Q \cap R) \leq Q$ .
Conjugation of any morphism in $G$ by a morphism in $F$ gives a morphism in $G$ , in the following sense: If $φ \in F$ and $α \in G$ are morphisms such that $φ \circ α \circ φ^{- 1}$ is well-defined and between two objects of $G$ (i.e., two subgroups of $Q$ ), then $φ \circ α \circ φ^{- 1} \in G$ .

References

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

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 A [[fusion subsystem]] <math>\mathcal{G}</math> of a [[fusion system]] <math>\mathcal{F}</math> on a [[group of prime power order]] <math>P</math> is termed a '''normal fusion subsystem''' if:
-* The subgroup <math>Q</math> of <math>P</math> for which <math>\mathcal{G}</math> is a [[fusion system]] is a [[strongly closed subgroup for a fusion system|strongly closed subgroup]] of <math>P</math>. In other words, for any <math>\varphi:R \to P</math>, <math>\varphi(Q \cap R) \le Q</math>.
+* The subgroup <math>Q</math> of <math>P</math> for which <math>\mathcal{G}</math> is a [[fusion system]] is a [[strongly closed subgroup for a fusion system|strongly closed subgroup]] of <math>P</math>. In other words, for any <math>\varphi:R \to P</math> with <math>\varphi \in \mathcal{F}</math>, <math>\varphi(Q \cap R) \le Q</math>.
 * Conjugation of any morphism in <math>\mathcal{G}</math> by a morphism in <math>\mathcal{F}</math> gives a morphism in <math>\mathcal{G}</math>, in the following sense: If <math>\varphi \in \mathcal{F}</math> and <math>\alpha \in \mathcal{G}</math> are morphisms such that <math>\varphi \circ \alpha \circ \varphi^{-1}</math> is well-defined and between two objects of <math>\mathcal{G}</math> (i.e., two subgroups of <math>Q</math>), then <math>\varphi\circ \alpha \circ \varphi^{-1} \in \mathcal{G}</math>.