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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.

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Direct finite group computations of cohomology of the finite groups of Lie type tend to be very sparse. The case $p=2$ has special interest for topologists and does provide some explicit results. More generally, some indirect results of interest have been found in recent decades by systematically comparing cohomology of the finite groups (in the defining characteristic $p$) with rational cohomology of the ambient algebraic groups. The main work in this direction is found in papers by some combination of Cline, Parshall, Scott, van der Kallen (see their joint 1977 paper in Invent. Math.), Friedlander. But a full description of the cohomology rings is elusive. Benson and Carlson have done a lot of work in the traditional setting as well. There are lots of papers, but not many definitive results.

For maximal elementary abelian $p$-subgroups, some of the work by Avrunin, Parshall, Scott would be relevant, as well as the older LMS Lecture Notes by Kleidman and Liebeck. Perhaps the most helpful source is the third volume of an ongoing series of books on the classification of finite simple groups. See especially section 3.3 and the summary table there for groups of Lie type:MR1490581 (98j:20011) 20D05 (20-02) Gorenstein, Daniel; Lyons, Richard (1-RTG); Solomon, Ronald (1-OHS) The classification of the finite simple groups. Number 3. Part I. Chapter A. Almost simple K-groups. Mathematical Surveys and Monographs, 40.3. American Mathematical Society, Providence, RI, 1998. xvi+419 pp. ISBN 0-8218-0391-3

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    $\begingroup$ The literature I was aware of five years ago is summarized in Chapters 14-15 of my 2006 book *Modular Representations of Finite Groups of Lie Type" (Cambridge Univ. Press), with lots of references. But the only really strong calculations occur for low degree cohomology with coefficients in trivial or other modules. $\endgroup$
    – Jim Humphreys
    Commented May 27, 2010 at 16:59

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