On the odd primary cohomology of higher projective planes

Let X be an n-fold loop space. Working with an auxiliary space p(p)(n)X analogous to the projective plane P(2)X, we show that the existence of certain Steenrod connections in H*(P(p)(n)X;F-p) (p odd) implies the vanishing of certain corresponding Dyer-Lashof operations in H*(X; F-p), and vice versa.