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 $\alpha(G)\subset {\mathbf G}\_{\mathbf a}^k\times\{0\}^{n-k}$?



Zariski-closed subgroups of ${\mathbf G}_{\mathbf a}^n$
The answer to your question is yes, assuming your algebraic groups are smooth (equivalently reduced, since your ground field $K$ is algebraically closed). 


Indeed, every smooth connected commutative $p$-torsion linear algebraic group over $K = \overline{K}$ of characteristic $p>0$ is a vector group (i.e., direct product of copies of $\mathbf{G}\_a$) -- one reference for this classical fact is Lemma B.1.10 in the book *Pseudo-reductive groups* -- and Proposition B.1.11 of the same book then gives exactly the affirmative answer to the question posed (with equality $\alpha(G) = \mathbf{G}\_a^k \times \{0\}^{n-k}$, same as inclusion for smooth connected $G$ by dimension reasons).


The entire Appendix B of that book (at least in its statements of results) is based on Tits' lectures at Yale in the late 1960's, so the result in question may be originally due to Tits but I am not sure.


