# Fusion system-equivalent finite groups

This article defines an equivalence relation over the collection of groups. View a complete list of equivalence relations on groups.

## Contents

## Definition

Suppose and are finite groups. We say that and are **fusion system-equivalent** if, for every prime number , the following two things are true:

- The -Sylow subgroup of is isomorphic to the -Sylow subgroup of . Note that there may be more than one -Sylow subgroup on each side, but by Sylow's theorem, they are conjugate, hence the criterion does not depend on the choice of Sylow subgroup.
- The fusion system induced by on its -Sylow subgroup is isomorphic (as a saturated fusion system) to the fusion system induced by on its -Sylow subgroup, where we talk of this "isomorphic" after identifying the two Sylow subgroups using any isomorphism that exists by condition (1).

## Relation with other equivalence relations

### Stronger equivalence relations

Relation | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

isomorphic groups | (obvious) | fusion system-equivalent not implies isomorphic |

## Facts

- Any two fusion system-equivalent groups have the same order.
- Finite nilpotent group implies every fusion system-equivalent group is isomorphic
- Fusion system-equivalence preserves perfectness