In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... This picture illustrates how the hours on a clock form a group under modular addition. ... In group theory, 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... 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. ... In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. ...

The commutator subgroup is important because it is the smallest normal subgroup such that the quotient group of the original group by this subgroup is abelian. So in some sense it provides a measure of how far the group is from being abelian; the larger the commutator subgroup is, the "less abelian" the group is. 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 intuitively a group that collapses the normal subgroup N to the identity element. ... In mathematics, an abelian group, also called a commutative group, is a group (G, * ) with the additional property that * commutes: for all a and b in G, a * b = b * a. ...

Definition

Given a group G the commutator subgroup [G,G] (also denoted G′ or G⁽¹⁾) of G is the subgroup generated by all the commutators ( [g,h]=g^-1h^-1gh ) of elements of G, that is 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. ...

This construction can be iterated:

G⁽⁰⁾: = G

A group with G⁽ⁿ⁾ = {e} for some n in N is called a solvable group. In the history of mathematics, the origins of group theory lie in the search for a proof of the general unsolvability of quintic and higher equations, finally realized by Galois theory. ...

An alternate construction can be iterated:

G'⁽⁰⁾: = G

A group with G'⁽ⁿ⁾ = {e} for some n in N is called a nilpotent group. In group theory, a nilpotent group is a group having a special property that makes it almost abelian, through repeated application of the commutator operation, [x,y] = x-1y-1xy. ...

The quotient group G / [G,G] is an abelian group called the abelianization of G or G made abelian. It is usually denoted by G^ab. The abelianization of G coincides with the first homology group of G. In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homeos = identical and logos = word) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ...

A group G is called a perfect group if [G,G] = G. Thus the abelianization of a perfect group is trivial. In mathematics, in the realm of group theory, a group is said to be perfect if it equals its own commutator subgroup. ...

Notes

In general the set of all commutators of the group is not a subgroup so we have to consider the subgroup generated by them. The smallest examples are two non-isomorphic groups of order 96. In each of these examples, the elements of the derived subgroup may be written as a product of no more than two commutators.

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.

Examples

• The commutator subgroup of the alternating group A₄ is the Klein four group
• The commutator subgroup of the symmetric group S_n is the alternating group A_n
• The commutator subgroup of the quaternion group Q = {1, −1, i, −i, j, −j, k, −k} is [Q,Q]={1, −1}.

In mathematics an alternating group is the group of even permutations of a finite set. ... In mathematics, the Klein four-group (or just Klein group or Vierergruppe, often symbolized by the letter V) is the group Z2 × Z2, the direct product of two copies of the cyclic group of order 2 (or any isomorphic variant). ... In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ... In mathematics an alternating group is the group of even permutations of a finite set. ... Cycle diagram of Q. Each color specifies a series of powers of any element connected to the identity element (1). ...

Properties

A group is abelian if and only if its commutator subgroup is the trivial group {e}. ↔ ⇔ ≡ logical symbols representing iff. ... The following list in mathematics contains the finite groups of small order up to group isomorphism. ...

Given a group G, a factor group G/N is abelian if and only if [G,G] ⊂ N.

If f : G → H is a group homomorphism, then f([G,G]) is a subgroup of [H,H], 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... Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ... In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ...

Applying this to endomorphisms of G, we find that [G,G] is a fully characteristic subgroup of G, and in particular a normal subgroup of G. (To reach the final conclusion, simply take conjugation with any particular g in G to be the automorphism in question. We see that g^-1[G,G]g = [G,G] for every g in G, and therefore that [G,G] is a normal subgroup of G. This is shown explicitly below). In mathematics, a characteristic subgroup of a group G is a subgroup H that is invariant under each automorphism of G. That is, if φ : G → G is a group automorphism (a bijective homomorphism from the group G to itself), then for every x in H we have φ(x) ∈ H: It... 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, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. ...

Universal property

The commutator subgroup satisfies the following universal property: 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. ...

Given a group G, the commutator subgroup [G,G] is the uniquely defined (up to isomorphism) subgroup of G so that given any homomorphism f : G → A from G to an abelian group A and the projection π : G → G/[G,G] then there exists a unique homomorphism s : G/[G,G] → A such that s o π = f

In other words, G^ab=G/[G,G] is the maximal abelian quotient of G. In mathematics, the term up to xxxx is used to describe a situation in which members of an equivalence class can be regarded as a single entity for some purpose. ... In mathematics, a projection is any one of several different types of functions, mappings, operations, or transformations, for example, the following: A set-theoretic operation typified by the jth projection map, written , that takes an element of the cartesian product to the value . ... In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is intuitively a group that collapses the normal subgroup N to the identity element. ...

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 mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ... In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another. ... A forgetful functor is a type of functor in mathematics. ...

Normality Property

The commutator subgroup is normal in G. That is, 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...

Proof:

Consider and arbitrary element . Then x = a ^{− 1}b ^{− 1}ab for some . Consider an arbitrary element . We want to show that . Consider the elements of G given by y = g ^{− 1}ag and z = g ^{− 1}bg. Then we can form which yields the element x ^{− 1}y ^{− 1}xy = g ^{− 1}a ^{− 1}gg ^{− 1}b ^{− 1}gg ^{− 1}agg ^{− 1}bg = g ^{− 1}a ^{− 1}b ^{− 1}abg which is in [G,G] by construction. Hence,

Q.E.D

See also

• solvable group
• nilpotent group