  In the integral cohomology ring of the classifying space of the projective
linear group $PGL_n$ (over $\mathbb{C}$), we find a collection of $p$-torsions
$y_{p,k}$ of degree $2(p^{k+1}+1)$ for any odd prime divisor $p$ of $n$, and
$k\geq 0$.
  If in addition, $p^2\nmid n$, there are $p$-torsion classes $\rho_{p,k}$ of
degree $p^{k+1}+1$ in the Chow ring of the classifying stack of $PGL_n$, such
that the cycle class map takes $\rho_{p,k}$ to $y_{p,k}$.
  We present an application of the above classes regarding Chern subrings.
