Specifically, let be a group. The group generated by the set of commutators of is called the derived group of . It is also called the commutator group of , though in general it is distinct from the set of commutators of . It is a normal subgroup of —in fact, it is a characteristic subgroup.
Evidently, if is a subgroup of , then is a subgroup of .
Proposition. Let and be groups, and be a group homomorphism. Then ; in fact, , so in particular, if is surjective, .
Proof. If is a commutator of , then is a commutator of ; thus . Suppose are points in the image of under ; let be elements of such that . Then . Hence .
Corollary 1. The derived group is a characteristic subgroup of . In particular, it is normal.
Corollary 2. The quotient group is commutative. Let be the canonical homomorphism from to . Let be an abelian group. Then every homomorphism can be expressed uniquely as , where is a homomorphism.
Proof. Note that ; thus is abelian. If is a homomorphism, then . Thus the realtion is compatible with equivalence mod .
Corollary 3. Let be a subgroup of . Then if and only if is normal in and is commutative.
Proof. If contains , then is normal in , since every subgroup of an abelian group is normal; hence is normal, and is isomorphic to , which is commutative. The converse follows from the previous corollary.
Corollary 4. Let be a generating subset of . Then is the normal subgroup generated by the set of commutators of elements of .
Proof. Let be the normal subgroup generated by the commutators of elements of . By definition, . On the other hand, the set generates the group ; since the elements of commute, is abelian, and hence .