Exotic fusion system

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Template:Fusion system property

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.

Definition

A saturated fusion system \mathcal{F} on a group of prime power order P is termed an exotic fusion system if there is no finite group G containing P as a Sylow subgroup for which \mathcal{F} equals the fusion system induced by G on P.

Facts

Measuring exoticity

  • Exoticity index provides a quantitative measurement of the exoticity of a saturated fusion system. It is nonzero iff the fusion system is exotic.

Examples