Homotopy commutativity of $H$-spaces with finitely generated cohomology

We show that a simply connected homotopy associative and homotopy commutative mod $3$ $H$-space with finitely generated mod $3$ cohomology is homotopy equivalent to a finite product of $K({\mathbb Z},2)$, $Sp(2)$, the three-connected cover $Sp(2)\langle 3\rangle$and the homotopy fiber $Sp(2)\langle 3;3^i\rangle$of the map $[3^i]:Sp(2)\to K({\mathbb Z},3)$ for $i\ge 1$. Our result also shows that a connected $C_p$-space in the sense of Sugawara with finitely generated mod $p$ cohomology has the homotopy type of a finite product of $K({\mathbb Z},1)$, $K({\mathbb Z},2)$ and $K({\mathbb Z}/p^i,1)$ for $i\ge 1$.