Question

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.

Answer 1

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.