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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 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