The derived group, in a sense, gives a measure of how far *G* is from being abelian; the larger *G*^{1}, the "less abelian" *G* is. In particular, *G*^{1} is equal to {1} if and only if the group *G* is abelian.

If *f* : *G* `->` *H* is a group homomorphism, then *f*(*G*^{1}) is a subset of *H*^{1}, because *f* maps commutators to commutators. This implies that the operation of forming derived groups is a functor from the category of groups to the category of groups.

Applying this to endomorphisms *f*, we find that *G*^{1} is a fully characteristic subgroup of *G*, and in particular a normal subgroup of *G*. The quotient *G*/*G*^{1}
is an abelian group sometimes called *G* **made abelian**, or the **abelianization** of *G*. In a sense, it is the abelian group that's "closest" to *G*, which can be expressed by the following universal property: if *p* : *G* `->` *G*/*G*^{1} is the canonical projection, and *f* : *G* `->` *A* is any homomorphism from *G* to an *abelian* group *A*, then there exists exactly one homomorphism *s* : *G*/*G*^{1} `->` *A* such that *s* o *p* = *f*. In the language of category theory: the functor which assigns to every group its abelianization is left adjoint to the forgetful functor which assigns to every abelian group its underlying group.

In particular, a quotient *G*/*N* of *G* is abelian if and only if *N* includes *G*^{1}.

A group is called **perfect** if it is equal to its derived group.