Is the affine space determined by its automorphism group?

In this note we study the problem of characterizing the complex affine space ${\mathbb{A}}^n$ via its automorphism group. We prove the following. Let $X$ be an irreducible quasi-projective $n$-dimensional variety such that $\mathrm{A}\mathrm{u}\mathrm{t}(X)$ and $\mathrm{A}\mathrm{u}\mathrm{t}({\mathbb{A}}^n)$ are isomorphic as abstract groups. If $X$ is either quasi-affine and toric or $X$ is smooth with Euler characteristic $\chi (X)\mathrm{\ne }0$ and finite Picard group $\mathrm{P}\mathrm{i}\mathrm{c}(X)$, then $X$ is isomorphic to ${\mathbb{A}}^n$. The main ingredient is the following result. Let $X$ be a smooth irreducible quasi-projective variety of dimension $n$ with finite $\mathrm{P}\mathrm{i}\mathrm{c}(X)$. If $X$ admits a faithful $(\mathbb{Z}\mathrm{/}p\mathbb{Z}{)}^n$-action for a prime $p$ and $\chi (X)$ is not divisible by $p$, then the identity component of the centralizer ${\mathrm{C}\mathrm{e}\mathrm{n}\mathrm{t}}_{\mathrm{A}\mathrm{u}\mathrm{t}(X)}((\mathbb{Z}\mathrm{/}p\mathbb{Z}{)}^n)$ is a torus.