# Finitely many commutators implies finite derived subgroup

This article gives a proof/explanation of the equivalence of multiple definitions for the term group with finite derived subgroup

View a complete list of pages giving proofs of equivalence of definitions

## Contents

## Statement

Suppose is a group such that the set of commutators in is a finite set. Then, the derived subgroup of (i.e., the subgroup of generated by the set of commutators) is a finite group, and hence, is a group with finite derived subgroup.

## Caveats

Note that this statement is *not* saying that if the derived subgroup is finitely generated, then it is finite. That statement is in fact false -- the derived subgroup of the infinite dihedral group is an infinite cyclic group. The statement is really about a very specific choice of generating set for the derived subgroup, namely, the set of *all* commutators.

## Facts used

- Finitely generated and FC implies FZ: The relevant part is that any finitely generated group that is also a FC-group (every conjugacy class is finite) is a FZ-group (the center has finite index).
- FZ implies finite derived subgroup (this result is also called the Schur-Baer theorem).

## Proof

**Given**: A group , with only finitely many elements that are commutators, say (note that the s are unique, but the are not uniquely determined).

**To prove**: The derived subgroup of is finite.

**Proof**:

Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|

1 | Let . Then, is a finitely generated subgroup of whose set of commutators is the same as the set of commutators of , and it is a finite set. | The whole given statement | is finitely generated by definition. Since , every commutator of elements of is a commutator of elements of . The other direction is obvious from the given: we've already written every commutator of elements of as a commutator of elements of . | ||

2 | is a FC-group, i.e., the size of the conjugacy class of every element is finite. | For any , we have . Thus, every conjugate of is a commutator of elements of times the element . Step (1) tells us that the set of commutators of elements of is finite, so the set of conjugates of is the image of a finite set and hence finite. | |||

3 | is a FZ-group, i.e., the center of has finite index in . | Fact (1) | Steps (1), (2) | By Step (1), is finitely generated. By Step (2), is a FC-group. Thus, by Fact (1), is a FZ-group. | |

4 | The derived subgroup is finite. | Fact (2) | Step (3) | Step-fact combination direct. | |

5 | is finite. | Steps (1), (4) | Step (1) says that . Combining with Step (4) gives the result. |

## References

- Math Stackexchange question about the subject. The proof presented on this page is an adaptation of that proof.