This is not so easy, but relies on a well-known structure theorem for connected group schemes over a perfect field.

Lemma 1. A finitely presented morphism $Y \to X$ of schemes is unramified if and only if $Y_x \to x$ is unramified for all $x \in X$.

Proof. See [EGA IV$_4$, Cor. 17.4.2]. $\square$

Thus, we may reduce to the case $R = k$ for $k$ a field, and by flat descent to the case where $k$ is algebraically closed. Then unramified for a finite extension means (geometrically) reduced.

Lemma 2. A finite group scheme over an algebraically closed field is a semidirect product of an étale group scheme and a connected group scheme.

Proof. See for example [Wat, §6.8]. The étale part is $\pi_0(G)$ and the connected part is $G^0$. $\square$

Theorem. If $G$ is a geometrically connected finite group scheme over a perfect field $k$ of characteristic $p > 0$, then $\Gamma(G,\mathcal O_G)$ is isomorphic to $k[X_1,\ldots,X_n]/(X_1^{p^{e_1}},\ldots,X_n^{p^{e_n}})$ for some $e_1,\ldots, e_n \in \mathbf Z_{>0}$.

Proof. See for example [Wat, §14.4]. I don't know a quick summary of why this is supposed to be true (I would be interested if someone does), but the proof is not that hard. $\square$

Theorem. If $G$ is group scheme over a field of characteristic $0$, then $G$ is geometrically reduced.

Proof. See for example [Wat, §11.4]. $\square$

In particular, if the rank of $G$ not divisible by $p$, then $G^0$ has to be trivial and $G$ is étale. $\square$

References.

[EGA IV$_4$] A. Grothendieck, Éléments de géométrie algébrique. IV: Étude locale des schémas et des morphismes de schémas (Quatrième partie).. Publ. Math., Inst. Hautes Étud. Sci. 32, p. 1-361 (1967). ZBL0153.22301.

[Wat] W.C. Waterhouse, Introduction to affine group schemes. Graduate Texts in Mathematics 66 (1979). ZBL0442.14017.

This is not so easy, but relies on a well-known structure theorem for connected group schemes over a perfect field.

Lemma 1. A finitely presented morphism $Y \to X$ of schemes is unramified if and only if $Y_x \to x$ is unramified for all $x \in X$.

Proof. See [EGA IV$_4$, Cor. 17.4.2]. $\square$

Thus, we may reduce to the case $R = k$ for $k$ a field, and by flat descent to the case where $k$ is algebraically closed. Then unramified for a finite extension means (geometrically) reduced.

Lemma 2. A finite group scheme over an algebraically closed field is a semidirect product of an étale group scheme and a connected group scheme.

Proof. See for example [Wat, §6.8]. The étale part is $\pi_0(G)$ and the connected part is $G^0$. $\square$

Theorem. If $G$ is a geometrically connected finite group scheme over a perfect field $k$ of characteristic $p > 0$, then $\Gamma(G,\mathcal O_G)$ is isomorphic to $k[X_1,\ldots,X_n]/(X_1^{p^{e_1}},\ldots,X_n^{p^{e_n}})$ for some $e_1,\ldots, e_n \in \mathbf Z_{>0}$.

Proof. See for example [Wat, §14.4]. I don't know a quick summary of why this is supposed to be true (I would be interested if someone does), but the proof is not that hard. $\square$

Theorem. If $G$ is group scheme over a field of characteristic $0$, then $G$ is geometrically reduced.

Proof. See for example [Wat, §11.4]. $\square$

In particular, if the rank of $G$ not divisible by $p$ (e.g. $p = 0$), then $G^0$ has to be trivial and $G$ is étale. $\square$

References.

[EGA IV$_4$] A. Grothendieck, Éléments de géométrie algébrique. IV: Étude locale des schémas et des morphismes de schémas (Quatrième partie).. Publ. Math., Inst. Hautes Étud. Sci. 32, p. 1-361 (1967). ZBL0153.22301.

[Wat] W.C. Waterhouse, Introduction to affine group schemes. Graduate Texts in Mathematics 66 (1979). ZBL0442.14017.

This is not so easy, but relies on a well-known structure theorem for connected group schemes over a perfect field.

Lemma 1. A finitely presented morphism $Y \to X$ of schemes is unramified if and only if $Y_x \to x$ is unramified for all $x \in X$.

Proof. See [EGA IV$_4$, Cor. 17.4.2]. $\square$

Thus, we may reduce to the case $R = k$ for $k$ a field, and by flat descent to the case where $k$ is algebraically closed. Then unramified for a finite extension means (geometrically) reduced.

Lemma 2. A finite group scheme over an algebraically closed field is a semidirect product of an étale group scheme and a connected group scheme.

Proof. See for example [Wat, §6.8]. The étale part is $\pi_0(G)$ and the connected part is $G^0$. $\square$

Theorem. If $G$ is a connected finite group scheme over a perfect field $k$ of characteristic $p > 0$, then $\Gamma(G,\mathcal O_G)$ is isomorphic to $k[X_1,\ldots,X_n]/(X_1^{p^{e_1}},\ldots,X_n^{p^{e_n}})$ for some $e_1,\ldots, e_n \in \mathbf Z_{>0}$.

Proof. See for example [Wat, §14.4]. I don't know a quick summary of why this is supposed to be true (I would be interested if someone does), but the proof is not that hard. $\square$

In particular, if the rank of $G$ is prime to $p$, then $G^0$ has to be trivial and $G$ is étale. $\square$

References.

[EGA IV$_4$] A. Grothendieck, Éléments de géométrie algébrique. IV: Étude locale des schémas et des morphismes de schémas (Quatrième partie).. Publ. Math., Inst. Hautes Étud. Sci. 32, p. 1-361 (1967). ZBL0153.22301.

[Wat] W.C. Waterhouse, Introduction to affine group schemes. Graduate Texts in Mathematics 66 (1979). ZBL0442.14017.

This is not so easy, but relies on a well-known structure theorem for connected group schemes over a perfect field.

Lemma 1. A finitely presented morphism $Y \to X$ of schemes is unramified if and only if $Y_x \to x$ is unramified for all $x \in X$.

Proof. See [EGA IV$_4$, Cor. 17.4.2]. $\square$

Thus, we may reduce to the case $R = k$ for $k$ a field, and by flat descent to the case where $k$ is algebraically closed. Then unramified for a finite extension means (geometrically) reduced.

Lemma 2. A finite group scheme over an algebraically closed field is a semidirect product of an étale group scheme and a connected group scheme.

Proof. See for example [Wat, §6.8]. The étale part is $\pi_0(G)$ and the connected part is $G^0$. $\square$

Theorem. If $G$ is a geometrically connected finite group scheme over a perfect field $k$ of characteristic $p > 0$, then $\Gamma(G,\mathcal O_G)$ is isomorphic to $k[X_1,\ldots,X_n]/(X_1^{p^{e_1}},\ldots,X_n^{p^{e_n}})$ for some $e_1,\ldots, e_n \in \mathbf Z_{>0}$.

Proof. See for example [Wat, §14.4]. I don't know a quick summary of why this is supposed to be true (I would be interested if someone does), but the proof is not that hard. $\square$

Theorem. If $G$ is group scheme over a field of characteristic $0$, then $G$ is geometrically reduced.

Proof. See for example [Wat, §11.4]. $\square$

In particular, if the rank of $G$ not divisible by $p$, then $G^0$ has to be trivial and $G$ is étale. $\square$

References.

[EGA IV$_4$] A. Grothendieck, Éléments de géométrie algébrique. IV: Étude locale des schémas et des morphismes de schémas (Quatrième partie).. Publ. Math., Inst. Hautes Étud. Sci. 32, p. 1-361 (1967). ZBL0153.22301.

[Wat] W.C. Waterhouse, Introduction to affine group schemes. Graduate Texts in Mathematics 66 (1979). ZBL0442.14017.