Fusion system-equivalence preserves perfectness

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Statement

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

Facts used

  1. Focal subgroup theorem

Proof

The key idea is to show that a finite group is perfect if and only if, for every prime, the focal subgroup for that prime (i.e., the focal subgroup of its Sylow subgroup) is the whole Sylow subgroup. This follows from Fact (1).