Some Facts
- Every cyclic group is locally cyclic, and every locally cyclic group is abelian.
- Every finitely-generated locally cyclic group is cyclic.
- Every subgroup and quotient group of a locally cyclic group is locally cyclic.
- A group is locally cyclic if and only if every pair of elements in the group generates a cyclic group.
- A group is locally cyclic if and only if its lattice of subgroups is distributive (Ore 1938).
- The torsion-free rank of a locally cyclic group is 0 or 1.
Read more about this topic: Locally Cyclic Group
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