# FZ-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

This article defines a term that has been used or referenced in a journal article or standard publication, but may not be generally accepted by the mathematical community as a standard term.[SHOW MORE]

## Definition

### Equivalent definitions in tabular format

No. | Shorthand | A group is termed a FZ-group if ... | A group is termed a FZ-group if ... |
---|---|---|---|

1 | center has finite index | its center is a subgroup of finite index in it. | the center is a subgroup of finite index, i.e., is finite. |

2 | inner automorphism group is finite | its inner automorphism group is a finite group. | the group of inner automorphisms (conjugations by elements of ) 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 and are both finite groups. |

4 | isoclinic to a finite group | it is isoclinic to a finite group. | there is a finite group such that and 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. | Finitely generated FZ-group|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) | Group with finite derived subgroup|FULL LIST, MORE INFO | ||

BFC-group | there is a common finite bound on the sizes of all conjugacy classes. | Group with finite derived subgroup|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.