# locally cyclic group

## Definition

A locally cyclic group is a group in which every finite subset generates a cyclic subgroup.

## Properties

From the definition we see that every finitely generated locally cyclic group (and, in particular, every finite locally cyclic group) is cyclic.

The following can all be shown to be equivalent for a group $G$:

• $G$ is locally cyclic.

• For all $a,b\in G$, the subgroup (http://planetmath.org/Subgroup) ${\left\langle a,b\right\rangle}$ is cyclic.

• $G$ is the union of a chain of cyclic subgroups.

• The lattice of subgroups of $G$ is distributive (http://planetmath.org/DistributiveLattice).

• $G$ embeds in $\mathbb{Q}$ or $\mathbb{Q}/\mathbb{Z}$.

• $G$ is isomorphic to a subgroup of a quotient (http://planetmath.org/QuotientGroup) of $\mathbb{Q}$.

• $G$ is involved in (http://planetmath.org/SectionOfAGroup) $\mathbb{Q}$.

From the last of these equivalent properties it is clear that every locally cyclic group is countable and abelian, and that subgroups and quotients of locally cyclic groups are locally cyclic.

 Title locally cyclic group Canonical name LocallyCyclicGroup Date of creation 2013-03-22 13:47:12 Last modified on 2013-03-22 13:47:12 Owner yark (2760) Last modified by yark (2760) Numerical id 24 Author yark (2760) Entry type Definition Classification msc 20K99 Classification msc 20E25 Synonym locally cyclic Synonym generalized cyclic group Synonym generalized cyclic Synonym generalised cyclic Synonym generalised cyclic group Related topic CyclicGroup Related topic AbelianGroup2 Related topic Group Related topic LocallyCalP