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
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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 $\phi :R\to P$, $\phi (Q\cap R)\le Q$.
• Conjugation of any morphism in $\mathcal{G}$ by a morphism in $\mathcal{F}$ gives a morphism in $\mathcal{G}$, in the following sense: If $\phi \in \mathcal{F}$ and $\alpha \in \mathcal{G}$ are morphisms such that $\phi \circ \alpha \circ {\phi }^{-1}$ is well-defined and between two objects of $\mathcal{G}$ (i.e., two subgroups of $Q$), then $\phi \circ \alpha \circ {\phi }^{-1}\in \mathcal{G}$.

References

• Introduction to Fusion Systems by Markus Linckelmann^{WeblinkMore info}