FC-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

Equivalent definitions in tabular format

No.	Shorthand	A group is termed a FC-group if ...	A group $G$ is termed a FC-group if ...
1	conjugacy classes are finite	every conjugacy class in it has finite size.	for every $x \in G$, the conjugacy class of $x$ in $G$ is finite.
2	element centralizers have finite index	the centralizer of any element is a subgroup of finite index.	for any $x \in G$, the index $[G:C_G(x)]$ of the centralizer $C_G(x)$ is finite.
3	finite subset centralizers have finite index	the centralizer of any finite subset is a subgroup of finite index.	for any finite subset $S \subseteq G$, the index $[G:C_G(S)]$ of the centralizer $C_G(S)$ is finite.
4	finitely generated subgroup centralizers have finite index	the centralizer of any subgroup generated by a finite subset is of finite index.	for any finitely generated subgroup $H$ of $G$, the index $[G:C_G(H)]$ is finite.

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
subgroup-closed group property	Yes		Suppose $G$ is a FC-group and $H$ is a subgroup of $G$. Then, $H$ is also a FC-group.
quotient-closed group property	Yes		Suppose $G$ is a FC-group and $H$ is a normal subgroup of $G$. Then, the quotient group $G/H$ is a FC-group.
finite direct product-closed group property	Yes		Suppose $G_1$ and $G_2$ are FC-groups. Then, so is the external direct product $G_1 \times G_2$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
finite group	has only finitely many elements	obvious	any infinite abelian group works as a counterexample.	FZ-group, Finitely generated FZ-group, Group with finite derived subgroup|FULL LIST, MORE INFO
abelian group	all conjugacy classes have size one	obvious	any finite non-abelian group works as a counterexample.	FZ-group, Group with finite derived subgroup|FULL LIST, MORE INFO
FZ-group	the center has finite index	FZ implies FC	FC not implies FZ	Group with finite derived subgroup|FULL LIST, MORE INFO
group with finite derived subgroup	the derived subgroup is finite	finite derived subgroup implies FC	FC not implies finite derived subgroup	|FULL LIST, MORE INFO
BFC-group	there is a common bound on the sizes of all conjugacy classes		FC not implies BFC	|FULL LIST, MORE INFO

Study of this notion

Mathematical subject classification

Under the Mathematical subject classification, the study of this notion comes under the class: 20F24

The subject classification 20F24 is used for FC-groups, and their generalizations.

External links

Definition links

• Wikipedia page
• Planetmath page