Group with finitely generated inner automorphism group

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 with finitely generated inner automorphism group if the inner automorphism group $\operatorname {Inn} (G)$ (which is isomorphic to the quotient group $G/Z(G)$ of $G$ by its center $Z(G)$) is a finitely generated group.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
FZ-group	inner automorphims group is finite			group with finitely generated inner automorphism group|FZ-group}}
finitely generated group	the group as a whole is finitely generated			|FULL LIST, MORE INFO
finite group	the group as a whole is finite	(via finitely generated, also via FZ)		|FULL LIST, MORE INFO
abelian group	any two elements commute, so the inner automorphism group is trivial	(via FZ)		|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
group in which every locally inner automorphism is inner		finitely generated inner automorphism group implies every locally inner automorphism is inner		|FULL LIST, MORE INFO