# Difference between revisions of "FC-group"

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{{group property}} | {{group property}} | ||

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==Definition== | ==Definition== | ||

− | === | + | ===Equivalent definitions in tabular format=== |

− | A [[ | + | {| class="sortable" border="1" |

+ | ! No. !! Shorthand !! A group is termed a FC-group if ... !! A group <math>G</math> is termed a FC-group if ... | ||

+ | |- | ||

+ | | 1 || conjugacy classes are finite || every [[conjugacy class]] in it has finite size. || for every <math>x \in G</math>, the conjugacy class of <math>x</math> in <math>G</math> is finite. | ||

+ | |- | ||

+ | | 2 || element centralizers have finite index || the [[centralizer]] of any element is a [[subgroup of finite index]]. || for any <math>x \in G</math>, the [[index of a subgroup|index]] <math>[G:C_G(x)]</math> of the [[centralizer]] <math>C_G(x)</math> is finite. | ||

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+ | | 3 || finite subset centralizers have finite index || the [[centralizer]] of any finite subset is a [[subgroup of finite index]]. || for any finite subset <math>S \subseteq G</math>, the [[index of a subgroup|index]] <math>[G:C_G(S)]</math> of the [[centralizer]] <math>C_G(S)</math> is finite. | ||

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+ | | 4 || finitely generated subgroup centralizers have finite index || the [[centralizer]] of any subgroup [[subgroup generated|generated by]] a finite subset is of finite index. || for any [[finitely generated group|finitely generated]] subgroup <math>H</math> of <math>G</math>, the [[index of a subgroup|index]] <math>[G:C_G(H)]</math> is finite. | ||

+ | |} | ||

− | + | ==Metaproperties== | |

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− | == | + | {| class="sortable" border="1" |

− | + | ! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols | |

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− | + | | [[satisfies metaproperty::subgroup-closed group property]] || Yes || || Suppose <math>G</math> is a FC-group and <math>H</math> is a subgroup of <math>G</math>. Then, <math>H</math> is also a FC-group. | |

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− | + | | [[satisfies metaproperty::quotient-closed group property]] || Yes || || Suppose <math>G</math> is a FC-group and <math>H</math> is a [[normal subgroup]] of <math>G</math>. Then, the quotient group <math>G/H</math> is a FC-group. | |

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+ | | [[satisfies metaproperty::finite direct product-closed group property]] || Yes || || Suppose <math>G_1</math> and <math>G_2</math> are FC-groups. Then, so is the [[external direct product]] <math>G_1 \times G_2</math>. | ||

+ | |} | ||

==Relation with other properties== | ==Relation with other properties== | ||

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===Stronger properties=== | ===Stronger properties=== | ||

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− | + | ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |

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− | + | | [[Weaker than::finite group]] || has only finitely many elements || obvious || any infinite abelian group works as a counterexample. || {{intermediate notions short|FC-group|finite group}} | |

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− | + | | [[Weaker than::abelian group]] || all conjugacy classes have size one || obvious || any finite non-abelian group works as a counterexample. || {{intermediate notions short|FC-group|abelian group}} | |

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− | + | | [[Weaker than::FZ-group]] || the center has finite index || [[FZ implies FC]] || [[FC not implies FZ]] || {{intermediate notions short|FC-group|FZ-group}} | |

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− | {{ | + | | [[Weaker than::group with finite derived subgroup]] || the [[derived subgroup]] is finite || [[finite derived subgroup implies FC]] || [[FC not implies finite derived subgroup]] || {{intermediate notions short|FC-group|group with finite derived subgroup}} |

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− | + | | [[Weaker than::BFC-group]] || there is a common bound on the sizes of all conjugacy classes || || [[FC not implies BFC]] || {{intermediate notions short|FC-group|BFC-group}} | |

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==Study of this notion== | ==Study of this notion== |

## Latest revision as of 04:19, 30 January 2013

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.