Hello,

in their book Cohomology of Finite Groups Adem and Milgram investigate the cohomology of the finite orthogonal and symplectic groups only in case $\mathbb{F}_2$.

Let $p$ be a prime dividing the order of $\text{O}_n(q)$, $\text{Sp}_n(q)$ and $q$ a prime power.

I am wondering if anything is known about $\text{H}^\ast(\text{O}_n(q),\mathbb{F}_p)$ or $\text{H}^\ast(\text{Sp}_n(q),\mathbb{F}_p)$.

I am also interested in the maximal elementary abelian $p$-subgroups of these groups.

In light of Quillen's stratification theorem these two questions are, of course, related to each other.

I would be grateful for any kind of information.