FZ-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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View other semistandard definitions in group theory

Definition

Equivalent definitions in tabular format

No.	Shorthand	A group is termed a FZ-group if ...	A group $G$ is termed a FZ-group if ...
1	center has finite index	its center is a subgroup of finite index in it.	the center $Z(G)$ is a subgroup of finite index, i.e., $G/Z(G)$ is finite.
2	inner automorphism group is finite	its inner automorphism group is a finite group.	the group $\operatorname {Inn} (G)$ of inner automorphisms (conjugations by elements of $G$) is finite.
3	inner automorphism group and derived subgroup both finite	both its inner automorphism group and its derived subgroup are finite groups.	the subgroups $\operatorname {Inn} (G)$ and $[G,G]$ are both finite groups.
4	isoclinic to a finite group	it is isoclinic to a finite group.	there is a finite group $K$ such that $G$ and $K$ are isoclinic groups.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
finite group	the whole group is finite		any infinite abelian group gives a counterexample.	|FULL LIST, MORE INFO
abelian group	any two elements commute		any finite non-abelian group gives a counterexample.	|FULL LIST, MORE INFO
finitely generated FZ-group				|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Stronger than:group with finite derived subgroup	finitely many commutators, or equivalently, the derived subgroup is finite.	FZ implies finite derived subgroup	finite derived subgroup not implies FZ	|FULL LIST, MORE INFO
group satisfying generalized subnormal join property	an arbitrary join of subnormal subgroups is subnormal.			|FULL LIST, MORE INFO
group satisfying subnormal join property	a join of finitely many subnormal subgroups is subnormal.			|FULL LIST, MORE INFO
FC-group	every conjugacy class is finite.	(via finite derived subgroup)		|FULL LIST, MORE INFO
BFC-group	there is a common finite bound on the sizes of all conjugacy classes.			|FULL LIST, MORE INFO
virtually abelian group	has an abelian subgroup of finite index.			|FULL LIST, MORE INFO

Metaproperties

Direct products

This group property is finite direct product-closed, viz the direct product of a finite collection of groups each having the property, also has the property
View other finite direct product-closed group properties

A direct product of finitely many FZ-groups is an FZ-group. This follows from the fact that the center is a direct product-preserved subgroup-defining function, that is, the center of a direct product of groups is the direct product of their centers.