On a theorem of Zelmanov (Q1818875)

The author gives a new (more direct and short) proof of the result of Zelmanov that a non-cyclic pro-\(p\) group \(G\) satisfying the Golod-Shafarevich inequality contains a nonabelian free pro-\(p\) subgroup. It is proved in addition that a quotient group of \(G\) contains a closed normal subgroup which is a Cartesian product of infinitely many finitely generated infinite pro-\(p\) groups, each of which is an epimorphic image of an open normal subgroup of \(G\).