# Locally free 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

This is a variation of free group|Find other variations of free group |

## Contents

## Definition

A group is termed a **locally free group** if every finitely generated subgroup of it is a free group.

## Formalisms

### In terms of the locally operator

This property is obtained by applying the locally operator to the property: free group

View other properties obtained by applying the locally operator

## Relation with other properties

### Stronger properties

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

locally cyclic aperiodic group | locally free and abelian; equivalently, locally cyclic and aperiodic | (by definition) | |FULL LIST, MORE INFO | |

free group | free on some generating set | freeness is subgroup-closed | locally free not implies free | Countably free group|FULL LIST, MORE INFO |

countably free group | every countable subgroup is free | follows from finitely generated implies countable | The group of rational numbers is locally free but not countably free | |FULL LIST, MORE INFO |

### Weaker properties

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

locally residually finite group |