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Let $X$ a affine algebraic variety. Then $X$ can have infinite automorphism group, for example $X=\mathbb{A}^1$. Let $G$ be a divisible abelian group. My question is about some condition in $G$ such that I can guarantee the existence of a algebraic variety $X$ such that $G$ is a subgroup of the isomorphism group of $X$.

More specifically, let $G_p$ the $p$-primary component of $G$. I want to construct $X$ with a rational point $P$ such that the orbit of $P$ by $G_p$ is rational over $\mathbb{Z}_p$. I want to know if there is some bibliography about this problem.

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  • $\begingroup$ You're using "isomorphism" instead of "automorphism". The distinction between this two words is precious! Second, "has infinite automorphisms" is senseless. I don't guess if you mean "has an infinite automorphism group" or "has automorphisms of infinite order". $\endgroup$
    – YCor
    Commented Jan 3, 2018 at 2:44
  • $\begingroup$ "Divisible group": are you assuming abelian? What is the $p$-primary component? the set of torsion elements whose order is a power of $p$? $\endgroup$
    – YCor
    Commented Jan 3, 2018 at 2:46
  • $\begingroup$ I'm sorry I want to mean "automorphism". With respect to the second question in principle $G$ is abelian, and the $p$ primary component is the set of elements whose order is a power of $p$. I though in the problem. Let $\tau(G)$ the torsion group of $G$. Maybe if $G/\tau(G)= \mathbb{Q}^n$ then it quotient induce the space $\mathbb{A}^n$ and $\tau(G)$ be the galois group of the variety over $\mathbb{A}^n$. $\endgroup$
    – camilo
    Commented Jan 3, 2018 at 17:10
  • $\begingroup$ I edited to fix wrong wording (I expected you would do so). Still I don't really understand what is the question, if there is any. $\endgroup$
    – YCor
    Commented Jan 4, 2018 at 0:06
  • $\begingroup$ I'm sorry the question was about if there are some bibliography in it direction. $\endgroup$
    – camilo
    Commented Jan 4, 2018 at 1:02

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