# Locally cyclic group

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Revision as of 00:15, 6 January 2009 by Vipul (talk | contribs) (New page: {{group property}} ==Definition== A group is termed '''locally cyclic''' if it satisfies the following equivalent conditions: # Every finitely generated...)

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Contents

## Definition

A group is termed **locally cyclic** if it satisfies the following equivalent conditions:

- Every finitely generated subgroup of the group is cyclic.
- The subgroup generated by any two elements of the group is cyclic.
- Its lattice of subgroups is a distributive lattice. In other words, the operations of join of subgroups and intersection of subgroups distribute over each other.