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.

Equivalence of definitions

Implication direction	Proof
(1) implies (2)	Schreier's lemma gives that any subgroup of finite index in a finitely generated group is finitely generated, hence, the center is finitely generated.
(2) implies (1)	If the center is finitely generated, then so is the whole group, because the center has finite index, so we need to add in only finitely many elements to the generating set of the center to get a generating set of the whole group.
(1) implies (3)	follows from FZ implies finite derived subgroup (also known as the Schur-Baer theorem).
(3) implies (4)	follows from finite derived subgroup implies FC (straightforward).
(4) implies (1)	see finitely generated and FC implies FZ.