subgoups of locally cyclic groups are locally cyclic

Theorem 1.

A group $G$ is locally cyclic iff every subgroup $H\leq G$ 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 $\langle S\rangle$ generated by $S$ is a cyclic subgroup of $G$, by assumption. Since every element $a$ of $\langle S\rangle$ is a product of elements or inverses of elements of $S$, and $S$ is a subset of group $H$, $a\in H$. Hence $\langle S\rangle$ 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. ∎

Title subgoups of locally cyclic groups are locally cyclic SubgoupsOfLocallyCyclicGroupsAreLocallyCyclic 2013-03-22 17:14:46 2013-03-22 17:14:46 rspuzio (6075) rspuzio (6075) 12 rspuzio (6075) Theorem msc 20E25 msc 20K99