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
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]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 |
