Finite nilpotent group implies every fusion system-equivalent group is isomorphic

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Statement

Suppose G_1 and G_2 are fusion system-equivalent finite groups. Suppose further that G_1 is a finite nilpotent group. Then, G_1 and G_2 are isomorphic groups. In particular, G_2 is also a finite nilpotent group.

Proof

The idea is to note that a finite group is nilpotent if and only if the fusion system induced on every Sylow subgroup is the inner fusion system for that subgroup.