# Locally cyclic group

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

## 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.
• 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.
$\mu _{p^{\infty }}=\left\{\exp \left({\frac {2\pi im}{p^{k}}}\right):m,k\in \mathbb {Z} \right\}$ 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, +) is not locally cyclic—the subgroup generated by 1 and π consists of all numbers of the form a + bπ. This group is isomorphic to the direct sum Z + Z, and this group is not cyclic.