Let $G$ be a simple algebraic group over an algebraically closed field k of characteristic $p$. Let $G(\mathbb{F}_{p^r})$ be the corresponding finite group of Lie type. Is it true that every maximal elementary abelian $p$-subgroup of $G(\mathbb{F}_{p^r})$ can be written as $H(\mathbb{F}_{p^r})$ where $H$ is a commutative algebraic subgroup of $G$? If not, is it known to be true under any circumstance?
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1$\begingroup$ This is closely related to a couple of earlier questions raised here, including mathoverflow.net/questions/188092/abelian-p-subgroups-of-e-6q/…. Note the references to earlier work of Mal'cev and Vdovin, and the detailed discussion of $p$-rank for simple groups of Lie type in Number 3 (section 3.3) of The Classification of Finite Simple Groups by Gorenstein-Lyons-Solomon (AMS, 1998). $\endgroup$– Jim HumphreysCommented Oct 6, 2015 at 23:48
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$\begingroup$ I'm inclined to think that the answer is yes, because one can regard any finite subgroup as an algebraic subgroup... If you want $H$ to be connected, then things are much more difficult... $\endgroup$– Nick GillCommented Oct 9, 2015 at 8:30
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$\begingroup$ Thanks so much Jim and Nick! I just got the book you said and I am attacking it. $\endgroup$– user81153Commented Oct 14, 2015 at 18:05
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