This article is concerned with homotopy properties of H-spaces X that are
reflected in the module of indecomposables QH*(X; [ ]p).
It is shown that mod p H-spaces X of finite type with finite transcendence degree mod p cohomology and
locally finite QH*(X; [ ]p) are Bℤ/p-null spaces,
Eilenberg–MacLane spaces K(ℤˆp, 2),
K(ℤ/pr, 1), and extensions of those.
If we restrict attention to H-spaces with
noetherian mod p cohomology algebra, then we are left with finite mod p H-spaces
and Eilenberg–MacLane spaces.