subgoups of locally cyclic groups are locally cyclic

Theorem 1.

A group $G$ is locally cyclic iff every subgroup $H\mathrm{\le }G$ is locally cyclic.

Proof.

Let $G$ be a locally cyclic group and $H$ a subgroup of $G$. Let $S$ be a finite subset of $H$. Then the group $⟨S⟩$ generated by $S$ is a cyclic subgroup of $G$, by assumption. Since every element $a$ of $⟨S⟩$ is a product of elements or inverses of elements of $S$, and $S$ is a subset of group $H$, $a\in H$. Hence $⟨S⟩$ is a cyclic subgroup of $H$, so $H$ is locally cyclic.

Conversely, suppose for every subgroup of $G$ is locally cyclic. Let $H$ be a subgroup generated by a finite subset of $G$. Since $H$ is locally cyclic, and $H$ itself is finitely generated, $H$ is cyclic, and therefore $G$ is locally cyclic.∎