Note on the Serre conjecture theorem of McGibbon and Neisendorfer (Q1313932)

Let \(X\) be a space so that it has a nontrivial, finitely generated, \(Z_ p\) homology. If \(X\) is simply connected, Neisendorfer and McGibbon have shown that infinitely many homotopy groups contain a subgroup of order \(p\). (This is Serre's conjecture.) If \(X\) is not simply connected, the conclusions do not hold. The author investigates conditions on \(\pi_ 1 (X)\) so that the conclusions still hold. They center around various concepts of nilpotence. As an application, the following interesting corollary is proved: Any connected finite \(H\)-space \(X\) has the homotopy type of a point or a torus or else \(\pi_ n (X)\) contains a subgroup of order \(p\) for infinitely many primes.