Lemma 70 ([Intersections retaining a perfect image]). Let f:M→H_0 be a homomorphism of finite groups, let P≤H_0 be perfect, and let B_1,…,B_s M , with s≥1 , satisfy P≤f(B_i) for every i . Then P≤f(⋂_ i=1 ^s B_i).