Fusion system-equivalence preserves perfectness
From Groupprops
Statement
Suppose and
are fusion system-equivalent finite groups. Suppose further that
is a perfect group. Then,
is also a perfect group.
Facts used
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).