FC-group
From Groupprops
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
Contents
Definition
Equivalent definitions in tabular format
No. | Shorthand | A group is termed a FC-group if ... | A group ![]() |
---|---|---|---|
1 | conjugacy classes are finite | every conjugacy class in it has finite size. | for every ![]() ![]() ![]() |
2 | element centralizers have finite index | the centralizer of any element is a subgroup of finite index. | for any ![]() ![]() ![]() |
3 | finite subset centralizers have finite index | the centralizer of any finite subset is a subgroup of finite index. | for any finite subset ![]() ![]() ![]() |
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 ![]() ![]() ![]() |
Metaproperties
Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|
subgroup-closed group property | Yes | Suppose ![]() ![]() ![]() ![]() | |
quotient-closed group property | Yes | Suppose ![]() ![]() ![]() ![]() | |
finite direct product-closed group property | Yes | Suppose ![]() ![]() ![]() |
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.