# Locally cyclic group

In group theory, 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.[1]
- 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 (Ore 1938).
- 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

- The additive group of rational numbers (
**Q**, +) is locally cyclic – any pair of rational numbers*a*/*b*and*c*/*d*is contained in the cyclic subgroup generated by 1/*bd*.[2] - The additive group of the dyadic rational numbers, the rational numbers of the form
*a*/2^{b}, is also locally cyclic – any pair of dyadic rational numbers*a*/2^{b}and*c*/2^{d}is contained in the cyclic subgroup generated by 1/2^{max(b,d)}. - Let
*p*be any prime, and let μ_{p∞}denote the set of all*p*th-power roots of unity in**C**, i.e.

- Then μ
_{p∞}is locally cyclic but not cyclic. This is the Prüfer*p*-group. The Prüfer 2-group is closely related to the dyadic rationals (it can be viewed as the dyadic rationals modulo 1).

## Examples of abelian groups that are not locally cyclic

- The additive group of real numbers (
**R**, +) is not locally cyclic—the subgroup generated by 1 and π consists of all numbers of the form*a*+*b*π. This group is isomorphic to the direct sum**Z**+**Z**, and this group is not cyclic.

## References

- Rose (2012), p. 54.
- Rose (2012), p. 52.

- Hall, Marshall, Jr. (1999), "19.2 Locally Cyclic Groups and Distributive Lattices",
*Theory of Groups*, American Mathematical Society, pp. 340–341, ISBN 978-0-8218-1967-8.

- Ore, Øystein (1938), "Structures and group theory. II" (PDF),
*Duke Mathematical Journal*,**4**(2): 247–269, doi:10.1215/S0012-7094-38-00419-3, MR 1546048.

- Rose, John S. (2012) [unabridged and unaltered republication of a work first published by the Cambridge University Press, Cambridge, England, in 1978].
*A Course on Group Theory*. Dover Publications. ISBN 0-486-68194-7.

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