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.
Contents
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, G/N is abelian if and only if N contains the commutator subgroup. 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 g and h of a group G, the commutator of g and h is
An element of G which is of the form
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,
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
It follows from the properties of commutators that any element of
for some natural number
so that
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 which is considerably stronger than normality.
The commutator subgroup can also be defined as the set of elements g of the group which have an expression as a product g = g1 g2 ... gk that can be rearranged to give the identity.
Derived series
This construction can be iterated:
The groups
is called the derived series. This should not be confused with the lower central series, whose terms are
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
The quotient
There is a useful categorical interpretation of the map
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
Classes of groups
A group G 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 G 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
A group with
A group with
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