Counterexamples to a conjecture of Adams

For any odd prime $p$ and any integer $n>0$ with $p^2\mathrm{\mid }n$, we show that the mod $p$ cohomology ring of the classifying space of the projective unitary group $PU(n)$ is not completely detected by elementary abelian $p$-subgroups, providing counterexamples to a conjecture due to J. F. Adams. We also give an application involving Milnor operations and Brown-Peterson cohomology.