On the cohomology of mod \(p\) homotopy commutative H-spaces (Q1204287)

It is shown that a \(p\)-local \(H\)-space with a homotopy commutative, homotopy associative multiplication and finitely many even degree algebra generators in the mod \(p\) cohomology can have even degree generators only in degrees \(2p^ k\) for \(k\geq 0\). Spaces which satisfy these hypotheses and do have generators in degrees \(2p^ k\) for \(0\leq k\leq 2\) are shown to exist.