  We study connected mod p finite A_p-spaces admitting AC_n-space structures
with n<p for an odd prime p. Our result shows that if n is greator than
(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 A_p-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 AC_n-space
structure compatible with the A_p-space structure.
