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.
[1][2] 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.
is equal to the identity element e if and only if
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:
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,
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.
[3] This motivates the definition of the commutator subgroup
(also called the derived subgroup, and denoted
) of G: it is the subgroup generated by all the commutators.
is of the form for some natural number
, the commutator subgroup is normal in G. For any homomorphism f: G → H, 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 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 = g1 g2 ... gk that can be rearranged to give the identity.
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
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.
There is a useful categorical interpretation of the map
there exists a unique homomorphism
As usual for objects defined by universal mapping properties, this shows the uniqueness of the abelianization
up to canonical isomorphism, whereas the explicit construction
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.
Equivalently, if and only if the group equals its abelianization.
for some n in N is called a solvable group; this is weaker than abelian, which is the case n = 1.
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).
has derived subgroup equal to itself,