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Can I have some examples of finite non-commutative connected group schemes over a field $k$?

I would like also to see some non-trivial torsors over a $k$-scheme $X$ under such group schemes. Thanks.

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Over a field of characteristic $p$, there is an obvious action of $\mathbb{G}_m$ on $\alpha_p$, and hence an action of $\mu_p$ on $\alpha_p$. The semi-direct product is a noncommutative connected group scheme of order $p^2$. – anon Jan 5 '12 at 9:16
up vote 6 down vote accepted

If $\mathrm{char}(k)=p>0$ and $G$ is a $k$-group scheme of finite type, the kernel of the relative frobenius $F_{G/k}:G\to G^{(p)}$ is a finite connected $k$-group scheme. It has the same Lie algebra as $G$, and in particular it is noncommutative if the Lie algebra is nonabelian, e.g. for $G=GL_{n,k}$, $n\geq2$.
If $G$ is smooth over $k$, then $F_{G/k}$ is faithfully flat, so we get an exact sequence (of pointed sets) $G(k)\to G^{(p)}(k)\to H^1(k, \ker(F_{G/k}))$ which may be used to get nontrivial torsors if $k$ is not perfect. For instance, if $G=GL_{n,k}$ as above, we have $G=G^{(p)}$ and $F_{G/k}$ acts on matrices by raising entries to the $p$-th power, so any $g\in GL_{n,k}$ whose entries are not all $p$-th powers gives rise to a nontrivial torsor.

Remark: the example given by anon as a comment to the question may be seen as the special case where $G$ is the natural semidirect product $\mathbb{G}_a \rtimes \mathbb{G}_m$, i.e. the group of affine automorphisms of the affine line.

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And one should note that Anon's example is a special case of this when $G$ is the semi-direct product of $\mathbb{G}_m$ and $\mathbb{G}_a$. – Moosbrugger Jan 5 '12 at 15:28
@Moosbrugger: yes, I was editing while you posted your comment! – Laurent Moret-Bailly Jan 5 '12 at 15:34
I think that if $G$ is regular but not smooth, then relative Frobenius is not faithfully flat. The point is that $G$ is geometrically reduced iff $F_{G/k}$ is schematically dominant. In other words : $G$ regular iff $F_G$ (absolute) faith. flat, while $G$ smooth iff $F_{G/k}$ faith. flat. – Matthieu Romagny Mar 17 '12 at 19:48
@Matthieu: corrigé, merci! – Laurent Moret-Bailly Mar 19 '12 at 13:33
Laurent, in fact I was somehow forgetting that since you are talking about a group scheme, it has a section, so that reduced implies geometrically reduced, and regular implies smooth. So you were right. Anyway, relative Frobenius exists not only for group schemes and it seems safer to say "$G/k$ smooth". – Matthieu Romagny Mar 19 '12 at 19:30

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