# Group with finite derived subgroup

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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## Definition

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

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

## 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 $H$ is a subgroup of a group $G$, and $G$ has finite derived subgroup, so does $H$.
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 $G$ has finite derived subgroup and $H$ is a normal subgroup of $G$. Then, the quotient group $G/H$ also has finite derived subgroup.
finite direct product-closed group property Yes follows from the derived subgroup operation commuting with direct products. Suppose $G_1$ and $G_2$ have finite derived subgroups. Then, so does the external direct product $G_1 \times G_2$.

## 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