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This question now has two sequels, Pointless groups II (to which @R.vanDobbendeBruyn gave a counterexample for an infinite, imperfect field) and Pointless groups III, both using revised wording suggested by @DanielLitt.

Fix a field $k$—not algebraically closed, and, in fact, this question is obviously of interest only if $k$ is finite. (EDIT: @R.vanDobbendeBruyn pointed out in the comments to Pointless groups II that my memory that the $k$-points are Zariski-dense in the infinite case is only guaranteed true for reductive groups.) Then "group" means "smooth, connected, affine algebraic group scheme of finite type over $k$".

Is it possible to have a (smooth, connected) group $G$, and a proper (smooth, connected) subgroup $H$, such that $G(k) = H(k)$? If so, then can this even happen for $H$ the trivial subgroup?

I have a small store of counterexamples showing that certain naïve statements about rational points of algebraic groups can fail for small fields $k$. (For example: $C_G(S)(k) \ne C_G(S(k))$ for $S$ a maximal torus in $\operatorname{SL}_2$ when $k$ has $2$ elements.) None of them provides an example of this sort of behaviour, but, then again, it is a very small testbed. If this question is too elementary for MO, then I have no objection to migrating it to MSE.

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Perhaps I'm missing something, but it seems to me that $k=\mathbb{F}_2$, $G=\mathbb{G}_{m, k}$ works, where $H$ is the trivial subgroup.

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    $\begingroup$ Ha! Well, that's embarrassing. My small collection of counterexamples should obviously be made bigger by including smaller groups. $\endgroup$
    – LSpice
    Commented Jun 10, 2022 at 20:45
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    $\begingroup$ @LSpice: I am curious if there are examples other than split tori over $\mathbb{F}_2$, with $H$ trivial (one can make lots of examples with $H$ non-trivial by taking products with this example). $\endgroup$
    – Daniel Litt
    Commented Jun 10, 2022 at 20:49
  • $\begingroup$ That's a good generalisation, that might well cover (in the sense of ruling out this sort of bad behaviour) the situation I have in mind. Do you think that that might be worth asking as a separate question? If so, then of course you would be welcome to ask it yourself, or I can if you prefer not to do so. $\endgroup$
    – LSpice
    Commented Jun 10, 2022 at 21:10
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    $\begingroup$ @LSpice you should feel free to ask it! I think it’s plausible these are the only examples and I have some idea for an argument, but there are some details to check. $\endgroup$
    – Daniel Litt
    Commented Jun 10, 2022 at 21:14
  • $\begingroup$ Done. Thanks! $\endgroup$
    – LSpice
    Commented Jun 10, 2022 at 22:58

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