Exotic fusion system

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

 * Every saturated fusion system on a finite p-group is induced by a possibly infinite group containing it as a Sylow subgroup
 * Every fusion system on a finite p-group is induced by a finite group containing it: Note that the finite group need not contain it as a Sylow subgroup.

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

 * Ruiz-Viruel exotic fusion systems are exotic fusion systems on the extraspecial group of order $$7^3 = 343$$ (i.e., the group unitriangular matrix group:UT(3,7)). This is the only group of prime-cube order that has an exotic fusion system.