Every saturated fusion system on a finite p-group is induced by a finite group containing it
Suppose is a group of prime power order with underlying prime . Suppose is a saturated fusion system on . Then there exists a finite group containing such that the fusion system induced by on is precisely . (Does the result hold for fusion systems that aren't saturated? No idea).
Note that need not contain as a -Sylow subgroup, even if is a saturated fusion system. If is a saturated fusion system and still cannot be induced from any finite group containing as a Sylow subgroup, then it is termed an exotic fusion system.