In finitely generated group, if $\{H_n\}_{n \in \mathbb{N}}$ is an infinite sequence of distinct cyclic group then $\cap{H_n}=\{e\}$.

Question

I think it is true. But I don't know the proof. And what about in countably generated group? More general, in which group it is true. Any kind of help will be appreciable.

Answer 1

Let $$G=\Bbb Z\rtimes\Bbb Z$$ as induced by the only non-trivial action, or if you prefer $$G=\langle \,a,b\mid ba=a^{-1}b\,\rangle,$$ or completely down to earth: $$G=\Bbb Z\times \Bbb Z$$ with multiplication rule $$ (a,b)\cdot (c,d):=(a+(-1)^bc,b+d).$$ Now let $H_n$ be generated by $(n,1)$. Then $(n,1)\cdot(n,1)=(0,2)\in H_n$ for all $n$.