# Group with finite derived subgroup

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

A group is said to have **finite derived subgroup** if it satisfies the following equivalent conditions:

- Its derived subgroup (i.e., its commutator subgroup with itself) is a finite group.
- The subset of elements of the group that can be written as commutators is a finite subset.

### Equivalence of definitions

- The direction (1) implies (2) is obvious.
- The direction (2) implies (1) is proved at finitely many commutators implies finite derived subgroup.

## Metaproperties

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

subgroup-closed group property | Yes | follows from the fact that derived subgroup of a subgroup is contained in derived subgroup of the whole group. | If is a subgroup of a group , and has finite derived subgroup, so does . |

quotient-closed group property | Yes | follows from the fact that derived subgroup of a quotient group is the image under the quotient map of the derived subgroup. | Suppose has finite derived subgroup and is a normal subgroup of . Then, the quotient group also has finite derived subgroup. |

finite direct product-closed group property | Yes | follows from the derived subgroup operation commuting with direct products. | Suppose and have finite derived subgroups. Then, so does 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 | the whole group has finitely many elements | (obvious) | any infinite abelian group gives a counterexample. | Finitely generated FZ-group|FULL LIST, MORE INFO |

abelian group | any two elements commute, or equivalently, the derived subgroup is trivial | (obvious) | any finite non-abelian group gives a counterexample. | |FULL LIST, MORE INFO |

FZ-group | the center has finite index, i.e., the inner automorphism group is finite. | FZ implies finite derived subgroup (this result is also called the Schur-Baer theorem) | finite derived subgroup not implies FZ | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

BFC-group | there is a common finite bound on the sizes of all conjugacy classes | this follows from the fact that every conjugacy class is contained in a coset of the derived subgroup | BFC not implies finite derived subgroup | |FULL LIST, MORE INFO |

FC-group | every conjugacy class is finite in size | this follows from the fact that size of conjugacy class is bounded by order of derived subgroup, which in turn is because every conjugacy class is contained in a coset of the derived subgroup. | FC not implies finite derived subgroup | |FULL LIST, MORE INFO |

locally FZ-group | every finitely generated subgroup is a FZ-group | (via FC) | (via FC) | |FULL LIST, MORE INFO |