# Locally cyclic group

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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.
^{[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.
^{[citation needed]}

## 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.- [math]\displaystyle{ \mu_{p^\infty} = \left\{\exp\left(\frac{2\pi im}{p^k}\right) : m, k \in \mathbb{Z}\right\} }[/math]

*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**, +); the subgroup generated by 1 and π (comprising all numbers of the form*a*+*b*π) is isomorphic to the direct sum**Z**+**Z**, which is not cyclic.

## References

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

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

- "Structures and group theory. II",
*Duke Mathematical Journal***4**(2): 247–269, 1938, doi:10.1215/S0012-7094-38-00419-3, http://dml.cz/bitstream/handle/10338.dmlcz/100155/CzechMathJ_05-1955-3_8.pdf.

- Rose, John S. (2012).
*A Course on Group Theory*. Dover Publications. ISBN 978-0-486-68194-8.

Original source: https://en.wikipedia.org/wiki/Locally cyclic group.
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