Normal fusion subsystem

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 ${\mathcal {G}}$ of a fusion system ${\mathcal {F}}$ on a group of prime power order $P$ is termed a normal fusion subsystem if:

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