Exotic fusion system

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

• 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.