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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3$\begingroup$ 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). $\endgroup$– YCorCommented Feb 20 at 19:32
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$\begingroup$ 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
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$\begingroup$ Exercise: embed this into the profinite group $\mathrm{GL}_2(\mathbf{Z}_3)$. $\endgroup$– YCorCommented Feb 21 at 11:33
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