Let $k$ be an algebraically closed field of characteristic $p>0$, and let $G$ be a reductive group defined over $\mathbb{F}\_p$. For any $d\in\mathbb{Z}^+$, let $C\_d(G)$ be the set of conjugacy classes of elementary abelian $p$-subgroups of $G(\mathbb{F}\_{q})$ of maximal rank, where $q=p^d$.



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> Is $S(G)=\displaystyle\sup\_{d\in\mathbb{Z}^+}\{|C\_d(G)|\}$ finite?
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A bit of motivation for my question:


For $n=2m$, the largest rank of an elementary abelian $p$-subgroup of $G=\mathrm{SL}\_n(\mathbb{F}\_q)$ is $dm^2$, and all such subgroups are conjugate to the upper right $m\times m$ block. In this case, we see that $|C\_d(G)|=1$ for all $d$. For $n=2m+1\ge 5$, the largest rank of an elementary abelian $p$-subgroup of $G=\mathrm{SL}\_n(\mathbb{F}\_q)$ is $dm(m+1)$, and all such subgroups are conjugate to either the upper right $(m+1)\times m$ block or the upper right $m\times(m+1)$ block, so that $|C\_d(G)|=2$ for all $d$.


In both examples, not only is $S(G)$ finite, but $|C\_d(G)|$ is independent of $d$. Perhaps this stronger statement is true. If not, I'd be interested to see a counter-example, that is, a reductive group $G$ and positive integers $d$ and $d'$ such that $|C\_d(G)|\ne C\_{d'}(G)|$.


I am fairly certain that $S(G)$ is finite for all simple groups of type $A$, $B,$ $C,$ and $D$. I'm less certain, but hopeful, that $S(G)$ is finite for the exceptional simple groups, and I have very little idea of what is happening in the general reductive case.



Elementary abelian $p$-subgroups of maximal rank in finite groups of Lie type
You might find it useful to look at some papers by Evgenii Vdovin. He has a bunch of papers discussing the maximal *orders* of nilpotent and abelian subgroups of finite groups of Lie type. 


Along the way he also calculates the $p$-rank of the finite groups of Lie type. In the process (I belive) he displays an elementary $p$-group of maximal rank inside each finite group of Lie type and I guess that the proof that such a group is maximal may well contain the ingredients to answer the question that you ask. (I don't think he explicitly answers your question anywhere unfortunately.)


Perhaps the most relevant paper in the series is this:



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> *Large abelian unipotent subgroups of finite Chevalley groups*. Algebra Logika 40 (2001), no. 5, 523--544, 624; translation in
>  Algebra Logic 40 (2001), no. 5, 292–305
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You can get hold of this paper if you have access to Springer's collection. However a free version of his PhD thesis is [online](http://math.nsc.ru/~vdovin/evdavg.ps), and I believe that contains all of the relevant results.


