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 $G$:

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$G$ is locally cyclic.

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For all $a,b\in G$, the subgroup^{} (http://planetmath.org/Subgroup) $\u27e8a,b\u27e9$ is cyclic.

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

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

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

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

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$G$ 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  20130322 13:47:12 
Last modified on  20130322 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 