Locally cyclic group

In mathematics, a locally cyclic group is a group (G, *) in which every finitely generated subgroup is cyclic.

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.
 * Every homomorphic image 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.
 * The torsion-free rank of a locally cyclic group is 0 or 1.
 * The endomorphism ring of a locally cyclic group is commutative.

Examples of abelian groups that are not locally cyclic

 * The additive group of real numbers (R, +); the subgroup generated by 1 and $\pi$ (comprising all numbers of the form a + bπ) is isomorphic to the direct sum Z + Z, which is not cyclic.