Finitely generated free group

This article defines a group property that is pivotal (i.e., important) among existing group properties
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VIEW RELATED: Group property implications | Group property non-implications | Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions |Group property dissatisfactions<random> @@@
RANDOM GROUP PROPERTY: <random>Simple group: A nontrivial group having no proper nontrivial normal subgroup.@@@Noetherian group: A group where every subgroup is finitely generated.@@@Complete group: A group for which the natural homomorphism to its automorphism group is an isomorphism, i.e., a centerless group for which every automorphism is inner.@@@SQ-universal group: A group for which every finitely generated group can be realized as a subquotient.@@@T-group: A group where any subnormal subgroup is normal, i.e., where normality is transitive. In general, normality is not transitive.@@@Locally finite group: A group in which every finitely generated subgroup is finite.@@@Group satisfying normalizer condition: A group with no proper self-normalizing subgroup.</random></random>

This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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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)	|FULL LIST, MORE INFO
Hopfian group	every surjective endomorphism is an automorphism	finitely generated and free implies Hopfian		|FULL LIST, MORE INFO
Locally free group				|FULL LIST, MORE INFO