Locally cyclic group

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

Definition

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

  1. Every finitely generated subgroup of the group is cyclic.
  2. The subgroup generated by any two elements of the group is cyclic.
  3. 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.

Relation with other properties

Stronger properties

Weaker properties