Locally cyclic group

Template:No footnotes 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.Template:Sfnp
• 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 Template:Harv.
• The torsion-free rank of a locally cyclic group is 0 or 1.
• The endomorphism ring of a locally cyclic group is commutative.Template:Citation needed

Examples of abelian groups that are not locally cyclic

• The additive group of real numbers (R, +); the subgroup generated by 1 and Template:Pi (comprising all numbers of the form a + bTemplate:Pi) is isomorphic to the direct sum Z + Z, which is not cyclic.

References

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