# 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 propertiesVIEW 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 is termed a FC-group if ... |
---|---|---|---|

1 | conjugacy classes are finite | every conjugacy class in it has finite size. | for every , the conjugacy class of in is finite. |

2 | element centralizers have finite index | the centralizer of any element is a subgroup of finite index. | for any , the index of the centralizer 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 , the index of the centralizer 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 of , the index is finite. |

## Metaproperties

Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|

subgroup-closed group property | Yes | Suppose is a FC-group and is a subgroup of . Then, is also a FC-group. | |

quotient-closed group property | Yes | Suppose is a FC-group and is a normal subgroup of . Then, the quotient group is a FC-group. | |

finite direct product-closed group property | Yes | Suppose and are FC-groups. Then, so is the external direct product . |

## 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.