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单词 LocallyCyclicGroup
释义

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 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 thatevery locally cyclic group is countableMathworldPlanetmath and abelianMathworldPlanetmath,and that subgroups and quotientsof locally cyclic groups are locally cyclic.

Titlelocally cyclic group
Canonical nameLocallyCyclicGroup
Date of creation2013-03-22 13:47:12
Last modified on2013-03-22 13:47:12
Owneryark (2760)
Last modified byyark (2760)
Numerical id24
Authoryark (2760)
Entry typeDefinition
Classificationmsc 20K99
Classificationmsc 20E25
Synonymlocally cyclic
Synonymgeneralized cyclic group
Synonymgeneralized cyclic
Synonymgeneralised cyclic
Synonymgeneralised cyclic group
Related topicCyclicGroup
Related topicAbelianGroup2
Related topicGroup
Related topicLocallyCalP
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更新时间:2025/5/4 11:35:07