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.

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