# Is there an infinite group generated by the elements of two (or more, but finitely many) normal subgroups?

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.

• No, finite normal subgroups always generate a finite normal subgroup. – Derek Holt Oct 25 '18 at 16:41

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.