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Let $\mathbf G$ be a connected algebraic group defined over a field $\mathbb F_p$. If $q=p^n$, then the groups $\mathbf G^\prime (\mathbb F_q)$ and $\mathbf G (\mathbb F_q)^\prime$ are not always equal (for example, when $\mathbf G=PGL_k$).

  1. What is known about the factor-groups $A_q=\mathbf G^\prime (\mathbb F_q)/ \mathbf G (\mathbb F_q)^\prime$?

In the case $\mathbf G=PGL_2$, the groups $A_q$ are isomorphic.

  1. Is it true that for an arbitrary $\mathbf G$, there exists some kind of periodicity in the sequence $\{A_{p^n}\}_n$?

In fact, I am interested in the case when $\mathbf G$ is unipotent. So, if it helps, you can add this condition to both questions.

Added: In the response on a comment of Jim Humphreys:

Let $\pi$ be a finite $p$-group, $\mathbb F$ an algebraic closure of $\mathbb F_p$ and $I$ the augmentation ideal of $\mathbb F[\pi]$. We can see $\mathbf G=1+I$ as an algebraic group. Together with Javier García and Urban Jezernik I have just proved that for every $q$ $$\mathbf G^\prime(\mathbb F_q)/\mathbf G(\mathbb F_q)^\prime\cong B_0(\pi),$$ where $B_0(\pi)$ is the Bogomolov multiplier of $\pi$.

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    $\begingroup$ It would help to see a nontrivial example involving a unipotent $G$, since this case sometimes exhibits quite different behavior compared with groups which are close to simple. $\endgroup$ Feb 3, 2015 at 18:36
  • $\begingroup$ @Jim Humphreys: I have put an example. $\endgroup$ Feb 3, 2015 at 19:52

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