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 locally cyclic groups that are not cyclic
Examples of abelian groups that are not locally cyclic
The additive group of real numbers (R, +); the subgroup generated by 1 and (comprising all numbers of the form a + b) is isomorphic to the direct sum Z + Z, which is not cyclic.
References
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