Locally cyclic group

In group theory, a locally cyclic group is a group ("G", *) in which every finitely generated subgroup is cyclic.

ome 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.
*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.

Examples of locally cyclic groups that are not cyclic

*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".
*Let "p" be any prime, and let μ_"p"^∞ denote the set of all "p"th-power roots of unity in C, i.e.

:$mu\_\{p^\{infty\; =\; left\{\; expleft(frac\{2pi\; im\}\{p^\{k\; ight)\; :\; m,kinmathbb\{Z\}\; ight\}$

:Then μ_"p"^∞ is locally cyclic but not cyclic. This is the Prüfer "p"-group.

Examples of abelian groups that are not locally cyclic

*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" + "b"π. This group is isomorphic to the direct sum Z + Z, and this group is not cyclic.