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.

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

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