Locally Cyclic Group - Some Facts

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.

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