Locally cyclic group

In mathematics, a locally cyclic group is a group (G, *) in which every finitely generated subgroup is cyclic.

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 pth-power roots of unity in C, i.e.
$\mu _{p^{\infty }}=\left\{\exp \left({\frac {2\pi im}{p^{k}}}\right):m,k\in \mathbb {Z} \right\}$
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).