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It's perfectly easy to get an infinite group generated by the elements of a finite set of finite subgroups: take a free product.

But is it possible to have an infinite group be generated by the elements of a finite set of finite normal subgroups? If so, I'd be interested either in a general construction yielding such groups or an interesting example or two.

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    $\begingroup$ No, finite normal subgroups always generate a finite normal subgroup. $\endgroup$ – Derek Holt Oct 25 '18 at 16:41
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If $H$ and $K$ are subgroup of $G$ and if one of them is normal, then $HK$ is a subgroup of $G$. So if they are both finite, they generate a finite subgroup.

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