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.
• 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.
• 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).

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.

References

• 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", Duke Mathematical Journal, 4 (2): 247–269, doi:10.1215/S0012-7094-38-00419-3, MR 1546048.