# Elementary abelian $p$-subgroups of maximal rank in finite groups of Lie type

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 $q=p^d$.

Is $S(G)=\displaystyle\sup_{d\in\mathbb{Z}^+}\{|C_d(G)|\}$ finite?

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.

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:

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

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, and I believe that contains all of the relevant results.

I believe the answer to your question (including the stronger version) is yes, and it may be written down explicitly in the finite group literature. Note that in the defining characteristic $p$ for a finite group of Lie type, the root structure plays a major role here since $p$-subgroups are the same thing as unipotent subgroups.

A basic source for the detailed structure of Sylow $p$-subgroups in all cases is Number 3 in the treatise The Classification of Finite Simple Groups by Gorenstein-Lyons-Solomon (AMS, 1998). See in particular their section 3.3, where for instance they have a convenient summary of the $p$-ranks of the finite groups of Lie type: this is the largest rank of an elementary abelian $p$-subgroup, important in the study of complexity and support varieties.

I'd have to look again at the literature including G-L-S to sort out what is actually known about conjugacy classes of elementary abelian $p$-subgroups, but in any case it's probably essential to relate the question to the Lie-theoretic structure inherited by the finite groups. In any case, the almost-simple groups are the essential ones to consider.

ADDED: Concerning my last remark, keep in mind that a (connected) reductive algebraic group is the almost-direct product of various almost-simple groups and a central torus. So for conjugacy purposes in the finite groups you can focus just on the almost-simple case.

G-L-S is not easy to read, because their study of known finite simple groups requires building considerable notation and terminology. However, they do present an authoritative version (modulo their posted corrections) of the internal structure of finite groups of Lie type. This gets complicated by the fact that most work has to be done case-by-case, while there are untwisted (split, Chevalley) groups as well as twisted (Steinberg, Suzuki, Ree) groups to consider. All of these groups draw their structure from the root system of a simple algebraic group, however. So in spite of the case arguments there is a lot of unity in the results. For instance, a typical Sylow $p$-subgroup comes from a set of positive roots, while conjugacy of Sylow subgroups and conjugacy theorems in the algebraic group help to focus attention on fixed root data.

While G-L-S primarily focus on finding maximal elementary abelian $p$-subgroups and the resulting $p$-rank, their results imply a reasonable finite upper bound on the number of conjugacy classes in the finite group. Roughly speaking, this comes from a case-by-case study of the "abelian" sets of positive roots (for which the corresponding root groups commute). But there are many special cases, for instance when $p$ is 2 or 3. So the details are delicate. On the other hand, passing to larger finite fields of characteristic $p$ obviously increases the $p$-rank, but the underlying size of maximal abelian subsets of roots is unaffected by that. The Lie-theoretic structure is especially crucial for exceptional types such as $E_8$, where some of the ideas go back to the older work of Mal'cev.