# Exotic fusion system

From Groupprops

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 -group is a -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 on a group of prime power order is termed an **exotic fusion system** if there is no finite group containing as a Sylow subgroup for which equals the fusion system induced by on .

## 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 (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.