# Fusion system-equivalent finite groups

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

## Definition

Suppose $G_1$ and $G_2$ are finite groups. We say that $G_1$ and $G_2$ are fusion system-equivalent if, for every prime number $p$, the following two things are true:

1. The $p$-Sylow subgroup of $G_1$ is isomorphic to the $p$-Sylow subgroup of $G_2$. Note that there may be more than one $p$-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.
2. The fusion system induced by $G_1$ on its $p$-Sylow subgroup is isomorphic (as a saturated fusion system) to the fusion system induced by $G_2$ on its $p$-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