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 $\mathbf G=PGL_k$).
- 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.
- 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 $\pi$.
