We study connected mod p finite Ap–spaces admitting
ACn–space structures with n<p for an odd prime p.
Our result shows that if n>(p-1)/2, then the mod p Steenrod algebra acts
on the mod p cohomology of such a space in a systematic way. Moreover, we
consider Ap–spaces which are mod p homotopy equivalent to
product spaces of odd dimensional spheres. Then we determine the largest
integer n for which such a space admits an ACn–space
structure compatible with the Ap–space structure.