Finitely generated free group

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This article defines a group property that is pivotal (i.e., important) among existing group properties
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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:

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