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Let $\mathbb{Z}[1/10]$ be an abelian group by addition. Let $\mathbb{Z}^2$ act on it by automorphisms by $x\mapsto 2x$ and $x\mapsto 5x$. Is the corresponding semidirect product $\mathbb{Z}^2\ltimes \mathbb{Z}[1/10]$ residually finite?

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    $\begingroup$ Yes, every f.g. metabelian group is residually finite (and such a particular case is easy to deal by hand — i.e. finite quotients are easy to construct). $\endgroup$
    – YCor
    Commented Feb 20 at 19:32
  • $\begingroup$ Is there any importance of the numbers 2 and 5? I'm curious to know how they arose. $\endgroup$
    – David Roberts
    Commented Feb 21 at 5:51
  • $\begingroup$ Exercise: embed this into the profinite group $\mathrm{GL}_2(\mathbf{Z}_3)$. $\endgroup$
    – YCor
    Commented Feb 21 at 11:33

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