Commutator subgroup

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.
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. In other words, formula_1 is abelian if and only if formula_2 contains the commutator subgroup of formula_3. 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.
Commutators.
For elements formula_4 and formula_5 of a group "G", the commutator of formula_4 and formula_5 is formula_8. The commutator formula_9 is equal to the identity element "e" if and only if formula_10 , that is, if and only if formula_4 and formula_5 commute. In general, formula_13.
However, the notation is somewhat arbitrary and there is a non-equivalent variant definition for the commutator that has the inverses on the right hand side of the equation: formula_14 in which case formula_15 but instead formula_16.
An element of "G" of the form formula_9 for some "g" and "h" is called a commutator. The identity element "e" = ["e","e"] is always a commutator, and it is the only commutator if and only if "G" is abelian.
Here are some simple but useful commutator identities, true for any elements "s", "g", "h" of a group "G":
The first and second identities imply that the set of commutators in "G" is closed under inversion and conjugation. If in the third identity we take "H" = "G", we get that the set of commutators is stable under any endomorphism of "G". This is in fact a generalization of the second identity, since we can take "f" to be the conjugation automorphism on "G", formula_26, to get the second identity.
However, the product of two or more commutators need not be a commutator. A generic example is ["a","b"]["c","d"] in the free group on "a","b","c","d". It is known that the least order of a finite group for which there exists two commutators whose product is not a commutator is 96; in fact there are two nonisomorphic groups of order 96 with this property.
Definition.
This motivates the definition of the commutator subgroup formula_27 (also called the derived subgroup, and denoted formula_28 or formula_29) of "G": it is the subgroup generated by all the commutators.
It follows from the properties of commutators that any element of formula_27 is of the form
for some natural number formula_32, where the "g""i" and "h""i" are elements of "G". Moreover, since for any "s" in "G" we have formula_33, the commutator subgroup is normal in "G". For any homomorphism "f": "G" → "H",
so that formula_35.
This shows that the commutator subgroup can be viewed as a functor on the category of groups, some implications of which are explored below. Moreover, taking "G" = "H" it shows that the commutator subgroup is stable under every endomorphism of "G": that is, ["G","G"] is a fully characteristic subgroup of "G", a property considerably stronger than normality.
The commutator subgroup can also be defined as the set of elements "g" of the group that have an expression as a product "g" = "g"1 "g"2 ... "g""k" that can be rearranged to give the identity.
Derived series.
This construction can be iterated:
The groups formula_38 are called the second derived subgroup, third derived subgroup, and so forth, and the descending normal series
is called the derived series. This should not be confused with the lower central series, whose terms are formula_40.
For a finite group, the derived series terminates in a perfect group, which may or may not be trivial. For an infinite group, the derived series need not terminate at a finite stage, and one can continue it to infinite ordinal numbers via transfinite recursion, thereby obtaining the transfinite derived series, which eventually terminates at the perfect core of the group.
Abelianization.
Given a group formula_3, a quotient group formula_1 is abelian if and only if formula_43.
The quotient formula_44 is an abelian group called the abelianization of formula_3 or formula_3 made abelian. It is usually denoted by formula_47 or formula_48.
There is a useful categorical interpretation of the map formula_49. Namely formula_50 is universal for homomorphisms from formula_3 to an abelian group formula_52: for any abelian group formula_52 and homomorphism of groups formula_54 there exists a unique homomorphism formula_55 such that formula_56. As usual for objects defined by universal mapping properties, this shows the uniqueness of the abelianization formula_47 up to canonical isomorphism, whereas the explicit construction formula_58 shows existence.
The abelianization functor is the left adjoint of the inclusion functor from the category of abelian groups to the category of groups. The existence of the abelianization functor Grp → Ab makes the category Ab a reflective subcategory of the category of groups, defined as a full subcategory whose inclusion functor has a left adjoint. 
Another important interpretation of formula_47 is as formula_60, the first homology group of formula_3 with integral coefficients.
Classes of groups.
A group formula_3 is an abelian group if and only if the derived group is trivial: ["G","G"] = {"e"}. Equivalently, if and only if the group equals its abelianization. See above for the definition of a group's abelianization.
A group formula_3 is a perfect group if and only if the derived group equals the group itself: ["G","G"] = "G". Equivalently, if and only if the abelianization of the group is trivial. This is "opposite" to abelian.
A group with formula_64 for some "n" in N is called a solvable group; this is weaker than abelian, which is the case "n" = 1.
A group with formula_65 for all "n" in N is called a non-solvable group.
A group with formula_66 for some ordinal number, possibly infinite, is called a hypoabelian group; this is weaker than solvable, which is the case "α" is finite (a natural number).
Perfect group.
Whenever a group formula_3 has derived subgroup equal to itself, formula_68, it is called a perfect group. This includes non-abelian simple groups and the special linear groups formula_69 for a fixed field formula_70.
Examples.
Map from Out.
Since the derived subgroup is characteristic, any automorphism of "G" induces an automorphism of the abelianization. Since the abelianization is abelian, inner automorphisms act trivially, hence this yields a map