Finitely generated free group

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

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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