---
tag: RXWA8TF2RM
kind: concept
name: Locally cyclic group
sources:
  - wikipedia: en:249268@r1353640923
    locator: Glossary of group theory, section L
    note: adopted
    mention: wikipedia://en/249268#Q3NQ
    sections: [Summary]
inventory: [Y76BK3HDCK]
---

## Summary

A group is locally cyclic if every finitely generated subgroup is cyclic. Every cyclic group is locally cyclic, and every finitely-generated locally cyclic group is cyclic. Every locally cyclic group is abelian. Every subgroup, every quotient group and every homomorphic image of a locally cyclic group is locally cyclic.