# Fusion system

## Contents

## Definition

The term **fusion system** is used for some additional structure (a category structure) on a finite p-group satisfying some conditions. There are two conflicting definitions of fusion system in use.

### Definition followed by Broto-Levi-Oliver

Broto-Levi-Oliver, Kessar, and others use the following convention: they define a fusion system as a category on a finite p-group that contains the inner fusion system. However, **they do not include the two other axioms -- the Sylow condition and the extension axiom -- that form part of the saturated fusion system definition**.

The two definitions are presented and contrasted below for a group of prime power order, say a finite -group, for a prime , and a category on :

Condition name | Qualitative description of condition | Condition details | Part of the Linckelman definition of fusion system (same as the Broto-Levi-Oliver definition of saturated fusion system)? | Part of the Broto-Levi-Oliver definition of fusion system? |
---|---|---|---|---|

contains inner fusion system | This condition makes sure that the obvious homomorphisms are included |
For any subgroups , all injective homomorphisms from to that arise as restrictions of inner automorphisms of , are present in . In other words, the inner fusion system on is a subcategory of . | Yes | Yes |

Sylow axiom | This condition is a purely global condition, i.e., it gives information only about automorphisms at the global level |
The inner automorphisms of form a Sylow subgroup of the group of automorphisms of in . | Yes | No |

Extension axiom | This condition links up the local and the global | Call a subgroup of fully normalized by if for any where is isomorphism in the category . Also define, for any morphism in : Then the statement of the extension axiom is:Every morphism such that is fully -normalized, extends to a morphism . |
Yes | No |

### Definition followed by Linckelman

Fusion systems as defined by Linckelman are what we here call saturated fusion systems. These are fusion systems **that also satisfy the Sylow axiom and the extension axiom.**

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.

## Fusion systems induced by groups

Given any finite group and any -subgroup of , we can define the fusion system induced by on . A case of special interest is where is a -Sylow subgroup of , and in that case we talk of the fusion system induced by a finite group on its p-Sylow subgroup. The following are true for fusion systems induced by finite groups on -subgroups:

Condition on group or subgroup embedding | Condition on fusion system | Does the condition on the group/subgroup imply the condition on the fusion system induced? | Given a fusion system with the condition, can we always find a group with the condition that induces that fusion system? |
---|---|---|---|

any | any | Yes | May be, it's true for saturated fusion systems |

the subgroup is a Sylow subgroup | the fusion system is a saturated fusion system | Yes | No (saturated fusion systems for which this isn't true are termed exotic fusion systems) |

the whole group is a simple group and the subgroup is Sylow | the fusion system is a simple fusion system | Yes | No (?) Don't really know |