The Wonderful Lemma, that was first proved by Roussel and Rubio, is one of the most important tools in the proof of the Strong Perfect Graph Theorem. Here we give a short proof of this lemma.

A group epimorphism Q: G+ Q preserves perfect radicals if $PG = PQ, where PC, PQ is maximal perfect in G, Q respectively. The following two questions are considered. Question 1. If P,,(B) acts trivially on H,(F; E), does the tibration F + Es B then have ~,(P)(P~,(E)) = Pm,(B)? Question 2. If f: X + Y is a map of spaces which induces an isomorphism of homology groups, is rr,(X) * rr,( Y) n,( Y)/...

For perfect fields k satisfying [k ¯:k]>2, we construct new normal subgroups of the plane Cremona group and provide an elementary proof its non-simplicity, following melody recent by Blanc, Lamy Zimmermann that rank n over (subfields of) complex numbers is not simple for n≥3.