# Locally cyclic group

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/2b, is also locally cyclic – any pair of dyadic rational numbers a/2b and c/2d is contained in the cyclic subgroup generated by 1/2max(b,d).
• Let p be any prime, and let μp denote the set of all pth-power roots of unity in C, i.e.
$\displaystyle{ \mu_{p^\infty} = \left\{\exp\left(\frac{2\pi im}{p^k}\right) : m, k \in \mathbb{Z}\right\} }$
Then μp 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 + bπ) is isomorphic to the direct sum Z + Z, which is not cyclic.

## References

1. Rose (2012), p. 54.
2. 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 .