Locally cyclic group

In mathematics, 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.^[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.

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).

The additive group of real numbers (R, +); the subgroup generated by 1 and $π$ (comprising all numbers of the form a + b $π$ ) is isomorphic to the direct sum Z + 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.

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.

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