Locally cyclic group
In mathematics, a locally cyclic group is a group 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 ; the subgroup generated by 1 and is isomorphic to the direct [sum of groups|direct sum] Z + Z, which is not cyclic.