In mathematics, the derived group (or commutator subgroup) of a group G is the subgroup G¹ generated by all the commutators of elements of G; that is, G¹ = <[g,h] : g,h in G>. Mathematics is the study of quantity, structure, space and change. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group operation... In abstract algebra, a generating set of a group is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses. ... For an electrical switch that periodically reverses the current see commutator (electric) In mathematics the commutator of two elements g and h of a group G is the element g −1 h −1 gh, often denoted by [ g, h ]. It is equal to the groups identity if and only...

The commutator subgroup can also be defined as the set of elements g of the group which have an expression as a product g=g₁g₂...g_k that can be rearranged to give the identity.

Note that the set of all commutators of the group is, generally, not a group (in any interesting case). While clumsily defined, the commutator subgroup is important.

An abelian group has only trivial commutators. Hence its commutator subgroup is {1}. The converse is also (trivially) true.

The derived group, in a sense, gives a measure of how far G is from being abelian; the larger G¹, the "less abelian" G is. In particular, G¹ is equal to {1} if and only if the group G is abelian. A perfect group G is one with G¹ = G. In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ...

If f : G -> H is a group homomorphism, then f(G¹) is a subgroup of H¹, 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. Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element... In category theory, a functor is a special type of mapping between categories. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ...

Applying this to endomorphisms f, we find that G¹ is a fully characteristic subgroup of G, and in particular a normal subgroup of G. The quotient G/G¹ 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/G¹ 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/G¹ -> 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 abstract algebra, a characteristic subgroup of a group G is a subgroup H of G invariant under each automorphism of G. This means that if f : G → G is a group automorphism (a bijective homomorphism from the group G to itself), then for every x in H we have... In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are... In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is a group that intuitively collapses the normal subgroup N to the identity element. ... In various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ... The existence of many pairs of adjoint functors is a major observation of the branch of mathematics known as category theory. ...

In particular, a quotient G/N of G is abelian if and only if N includes G¹.