# Locally cyclic group

From Groupprops

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

### Symbol-free 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.
- It is isomorphic to a subquotient (i.e., a quotient group of a subgroup) of the group of rational numbers.
- 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.

### Equivalence of definitions

`Further information: Locally cyclic iff subquotient of rationals, Locally cyclic iff distributive lattice of subgroups`

(1) and (2) are clearly equivalent. For the equivalence of (1) and (2) with (3), refer locally cyclic iff subquotient of rationals. For the equivalence with (4), refer locally cyclic iff distributive lattice of subgroups.