  We study the mod p cohomology of the classifying space of the projective
unitary group PU(p). We first proof that old conjectures due to J.F. Adams, and
Kono and Yagita about the structure of the mod p cohomology of classifying
space of connected compact Lie groups held in the case of PU(p). Finally, we
proof that the classifying space of the projective unitary group PU(p) is
determined by its mod p cohomology as an unstable algebra over the Steenrod
algebra for p>3, completing previous works for the cases p=2,3.
