Locally cyclic group

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.
• 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 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.
• Let p be any prime, and let μ_p^∞ denote the set of all pth-power roots of unity in C, i.e.

  Examples of abelian groups that are not locally cyclic

• The additive group of real numbers 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.