Finitely generated FZ-group

This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: finitely generated group and FZ-group
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Definition

Equivalent definitions in tabular format

No.	Shorthand	A group is termed a finitely generated FZ-group if ...	A group $G$ is termed a finitely generated FZ-group if ...
1	it is finitely generated and FZ	it is both a finitely generated group and a FZ-group.	$G$ has a finite generating set and its inner automorphism group is a finite group.
2	FZ and center is finitely generated abelian	it is a FZ-group and its center is a finitely generated abelian group.	$G/Z(G)$ is a finite group and $Z(G)$ is a finitely generated abelian group.
3	finitely generated and finite derived subgroup	it is a finitely generated group and also a group with finite derived subgroup, i.e., its derived subgroup is a finite group.	$G$ has a finite generating set and the derived subgroup $G'$ is a finite group.
4	finitely generated and FC	it is a finitely generated group and a FC-group	$G$ has a finite generating set and every conjugacy class in $G$ is finite.