# 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 $\u27e8S\u27e9$ generated by $S$ is a cyclic subgroup of $G$, by assumption. Since every element $a$ of $\u27e8S\u27e9$ is a product of elements or inverses^{} of elements of $S$, and $S$ is a subset of group $H$, $a\in H$. Hence $\u27e8S\u27e9$ 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.
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Title | subgoups of locally cyclic groups are locally cyclic |
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Canonical name | SubgoupsOfLocallyCyclicGroupsAreLocallyCyclic |

Date of creation | 2013-03-22 17:14:46 |

Last modified on | 2013-03-22 17:14:46 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 12 |

Author | rspuzio (6075) |

Entry type | Theorem |

Classification | msc 20E25 |

Classification | msc 20K99 |