Let's work over an algebraically closed field $K$. A $1$-dimensional connected closed subgroup of ${\mathbf G}_{\mathbf a}^n$ is isomorphic to ${\mathbf G}_{\mathbf a}^1$. If $K$ has characteristic $0$, then a closed subgroup $G\subset{\mathbf G}_{\mathbf a}^n$ is simply a $K$-vector space since for any non zero $g\in G$, the line $L$ directed by $g$ intersets $G$ in infinitely many points, implying $L\subset G$.

Are there any references concerning the structure of closed subgroups of ${\mathbf G}_{\mathbf a}^n$ when $K$ has characteristic $p$ ?

For instance, if a closed connected subgroup $G\subset {\mathbf G}_{\mathbf a}^n$ has dimension $k$, does there exist a group automorphism $\alpha$ of ${\mathbf G}_{\mathbf a}^n$ such that $\alpha(G)\subset {\mathbf G}_{\mathbf a}^k\times\{0\}^{n-k}$?

Let's work over an algebraically closed field $K$. A $1$-dimensional Zariski-closed connected subgroup of ${\mathbf G}_{\mathbf a}^n$ is isomorphic to ${\mathbf G}_{\mathbf a}^1$. If $K$ has characteristic $0$, then a Zariski-closed subgroup $G\subset{\mathbf G}_{\mathbf a}^n$ is simply a $K$-vector space since for any non zero $g\in G$, the line $L$ directed by $g$ intersets $G$ in infinitely many points, implying $L\subset G$.

Are there any references concerning the structure of Zariski-closed subgroups of ${\mathbf G}_{\mathbf a}^n$ when $K$ has characteristic $p$ ?

For instance, if a Zariski-closed connected subgroup $G\subset {\mathbf G}_{\mathbf a}^n$ has dimension $k$, does there exist a group automorphism $\alpha$ of ${\mathbf G}_{\mathbf a}^n$ such that $\alpha(G)\subset {\mathbf G}_{\mathbf a}^k\times\{0\}^{n-k}$?

Let's work over an algebraically closed field $K$. A $1$-dimensional connected closed subgroup of ${\mathbf G}_{\mathbf a}^n$ is isomorphic to ${\mathbf G}_{\mathbf a}^1$. If $K$ has characteristic $0$, then a closed subgroup $G\subset{\mathbf G}_{\mathbf a}^n$ is simply a $K$-vector space since for any non zero $g\in G$, the line $L$ directed by $g$ intersets $G$ in infinitely many points, implying $L\subset G$.

Are there any references concerning the structure of closed subgroups of ${\mathbf G}_{\mathbf a}^n$ when $K$ has characteristic $p$ ?

For instance, if a closed connected subgroup $G\subset {\mathbf G}_{\mathbf a}^n$ has dimension $k$, does there exist an isomorphism $\alpha$ of ${\mathbf G}_{\mathbf a}^n$ such that $\alpha(G)\subset {\mathbf G}_{\mathbf a}^k\times\{0\}^{n-k}$?

Let's work over an algebraically closed field $K$. A $1$-dimensional connected closed subgroup of ${\mathbf G}_{\mathbf a}^n$ is isomorphic to ${\mathbf G}_{\mathbf a}^1$. If $K$ has characteristic $0$, then a closed subgroup $G\subset{\mathbf G}_{\mathbf a}^n$ is simply a $K$-vector space since for any non zero $g\in G$, the line $L$ directed by $g$ intersets $G$ in infinitely many points, implying $L\subset G$.

Are there any references concerning the structure of closed subgroups of ${\mathbf G}_{\mathbf a}^n$ when $K$ has characteristic $p$ ?

For instance, if a closed connected subgroup $G\subset {\mathbf G}_{\mathbf a}^n$ has dimension $k$, does there exist a group automorphism $\alpha$ of ${\mathbf G}_{\mathbf a}^n$ such that $\alpha(G)\subset {\mathbf G}_{\mathbf a}^k\times\{0\}^{n-k}$?

Let's work over an algebraically closed field $K$. A $1$-dimensional connected closed subgroup of ${\mathbf G}_{\mathbf a}^n$ is isomorphic to ${\mathbf G}_{\mathbf a}^1$. If $K$ has characteristic $0$, then a closed subgroup $G\subset{\mathbf G}_{\mathbf a}^n$ is simply a $K$-vector space since for any non zero $g\in G$, the line $L$ directed by $g$ intersets $G$ in infinitely many points, implying $L\subset G$.

Are there any references concerning the structure of closed subgroups of ${\mathbf G}_{\mathbf a}^n$ when $K$ has characteristic $p$ ?

For instance, if a closed connected subgroup $G\subset {\mathbf G}_{\mathbf a}^n$ has dimension $k$, does there exist an isomorphism of variety $\alpha$ of $K^n$ such that $\alpha(G)\subset {K}^{k}\times\{0\}^{n-k}$?

Let's work over an algebraically closed field $K$. A $1$-dimensional connected closed subgroup of ${\mathbf G}_{\mathbf a}^n$ is isomorphic to ${\mathbf G}_{\mathbf a}^1$. If $K$ has characteristic $0$, then a closed subgroup $G\subset{\mathbf G}_{\mathbf a}^n$ is simply a $K$-vector space since for any non zero $g\in G$, the line $L$ directed by $g$ intersets $G$ in infinitely many points, implying $L\subset G$.

Are there any references concerning the structure of closed subgroups of ${\mathbf G}_{\mathbf a}^n$ when $K$ has characteristic $p$ ?

For instance, if a closed connected subgroup $G\subset {\mathbf G}_{\mathbf a}^n$ has dimension $k$, does there exist an isomorphism $\alpha$ of ${\mathbf G}_{\mathbf a}^n$ such that $\alpha(G)\subset {\mathbf G}_{\mathbf a}^k\times\{0\}^{n-k}$?