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