Group in which every locally inner automorphism is inner

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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Definition

A group $G$ is termed a group in which every locally inner automorphism is inner if every locally inner automorphism of $G$ is an inner automorphism of $G$. Here, locally inner means that for any finite subset, its action on that finite subset agrees with some inner automorphism.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
group with finitely generated inner automorphism group	the inner automorphism group is a finitely generated group	finitely generated inner automorphism group implies every locally inner automorphism is inner		|FULL LIST, MORE INFO
FZ-group	the inner automorphism group is finite	(via finitely generated inner automorphism group)		Template:Intermediate notions sohrt
finitely generated group	the whole group is finitely generated, i.e., has a finite generating set	(via finitely generated inner automorphism group)		|FULL LIST, MORE INFO
finite group	the whole group is finite	(via finitely generated, also via FZ)		|FULL LIST, MORE INFO
abelian group	the whole group is abelian, so the inner automorphism group is trivial	(via FZ)		|FULL LIST, MORE INFO