Normal fusion subsystem

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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) \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 \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}.

References