In group theory, a locally cyclic group is a group (G, *) in which every finitely generated subgroup is cyclic.
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.
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.
μ
p
∞
=
{
exp
(
2
π
i
m
p
k
)
:
m
,
k
∈
Z
}
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).
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.
Locally cyclic group Wikipedia (Text) CC BY-SA