Locally cyclic group explained

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 (comprising all numbers of the form a + b) is isomorphic to the direct sum Z + Z, which is not cyclic.

References

• .
• .
• Book: Rose, John S.. A Course on Group Theory. 2012. Dover Publications. 978-0-486-68194-8. unabridged and unaltered republication of a work first published by the Cambridge University Press, Cambridge, England, in 1978.