# Finitely generated free group

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## Definition

### Symbol-free definition

A group is said to be a **finitely generated free group** if it satisfies the following equivalent conditions:

- It is finitely generated (that is, it has a finite generating set) and is also a free group
- It is has a finite freely generating set, viz it is freely generated by a finite set

### Definition with symbols

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Often, people mean **finitely generated free group** when they just say **free group**.

## Relation with other properties

### Weaker properties

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

Free group | free on a subset (need not be a finite subset) | |FULL LIST, MORE INFO | ||

Finitely generated residually finite group | finitely generated and residually finite | free implies residually finite | (finite nontrivial groups give obvious counterexamples) | Finitely presented conjugacy-separable group, Finitely presented residually finite group|FULL LIST, MORE INFO |

Hopfian group | every surjective endomorphism is an automorphism | finitely generated and free implies Hopfian | Finitely generated Hopfian group, Finitely generated residually finite group, Finitely presented conjugacy-separable group, Finitely presented residually finite group|FULL LIST, MORE INFO | |

Locally free group | |FULL LIST, MORE INFO |