Group with finite derived subgroup

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.

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.