Locally cyclic group

• 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==

{{unordered list : \mu_{p^\infty} = \left\{\exp\left(\frac{2\pi im}{p^k}\right) : m, k \in \mathbb{Z}\right\} 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 cyclic==

• The additive group of real numbers (R, +); the subgroup generated by 1 and (comprising all numbers of the form a + b) is isomorphic to the direct sum Z + Z, which is not cyclic. == See also ==