# 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.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

cyclic group | generated by one element | locally cyclic not implies cyclic | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

abelian group | any two elements commute | locally cyclic implies abelian | abelian not implies locally cyclic | Epabelian group|FULL LIST, MORE INFO |

group whose automorphism group is abelian | automorphism group is abelian | locally cyclic implies abelian automorphism group | abelian and abelian automorphism group not implies locally cyclic | |FULL LIST, MORE INFO |

group with at most n elements of order dividing n | |FULL LIST, MORE INFO | |||

group in which every finite subgroup is cyclic | every finite subgroup is cyclic | Group with at most n elements of order dividing n|FULL LIST, MORE INFO |

### Weaker properties conditional to nontriviality

The following properties are weaker if we restrict to nontrivial groups.

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

splitting-simple group | has no proper nontrivial complemented normal subgroup | |FULL LIST, MORE INFO | ||

directly indecomposable group | has no proper nontrivial direct factor | |FULL LIST, MORE INFO |