Homotopy commutativity of -spaces with finitely generated cohomology

We show that a simply connected homotopy associative and homotopy commutative mod -space with finitely generated mod cohomology is homotopy equivalent to a finite product of , , the three-connected cover and the homotopy fiber of the map for . Our result also shows that a connected -space in the sense of Sugawara with finitely generated mod cohomology has the homotopy type of a finite product of , and for .