locally cyclic group


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


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

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

  • G is locally cyclic.

  • For all a,bG, the subgroupMathworldPlanetmathPlanetmath (http://planetmath.org/Subgroup) a,b 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 or /.

  • G is isomorphicPlanetmathPlanetmathPlanetmath to a subgroup of a quotient (http://planetmath.org/QuotientGroup) of .

  • G is involved in (http://planetmath.org/SectionOfAGroup) .

From the last of these equivalent properties it is clear that every locally cyclic group is countableMathworldPlanetmath and abelianMathworldPlanetmath, 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