Examples of locally cyclic groups that are not cyclicedit
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.
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 cyclicedit
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 sumZ + Z, which is not cyclic.
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" (PDF), Duke Mathematical Journal, 4 (2): 247–269, doi:10.1215/S0012-7094-38-00419-3, MR 1546048.
Rose, John S. (2012) [unabridged and unaltered republication of a work first published by the Cambridge University Press, Cambridge, England, in 1978]. A Course on Group Theory. Dover Publications. ISBN 978-0-486-68194-8.