Commutator subgroup
Encyclopedia
In mathematics
, more specifically in abstract algebra
, the commutator subgroup or derived subgroup of a group
is the subgroup
generated
by all the commutator
s 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, 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.
of g and h is
. The commutator [g,h] is equal to the identity element e if and only if gh = hg, that is, if and only if g and h commute. In general, 
.
An element of G which is of the form [g,h] for some g and h is called a commutator. The identity element
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 under conjugation. If in the third identity we take
, 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 
.
However, the product of two or more commutators need not be a commutator. A generic example is
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.
by all the commutators.
It follows from the properties of commutators that any element of [G,G] is of the form
for some natural number n. Moreover, since
, the commutator subgroup is normal in G. For any homomorphism 
,
,
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
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 = g1g2...gk that can be rearranged to give the identity.
The groups
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
, not 
.
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 number
s via transfinite recursion, thereby obtaining the transfinite derived series, which eventually terminates at the perfect core
of the group.
The quotient
is an abelian group called the abelianization of G or G made abelian. It is usually denoted by Gab or Gab.
There is a useful categorical interpretation of the map
. Namely 
is universal for homomorphisms from
G to an abelian group H: for any abelian group H and homomorphism of groups
there exists a unique homomorphism 
such that 
. As usual
for objects defined by universal mapping properties, this shows the uniqueness of the
abelianization
up to canonical isomorphism, whereas the
explicit construction
shows existence.
The abelianization functor is the left adjoint
of the inclusion functor from the category of abelian groups to the category of groups.
Another important interpretation of
is as
, the first homology group of G with integral coefficients.
if and only if the derived group is trivial:
. 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:
. Equivalently, if and only if the abelianization of the group is trivial. This is "opposite" to abelian.
A group with
for some n in N is called a solvable group
; this is weaker than abelian, which is the case
.
A group with
for some ordinal number
, possibly infinite, is called a hypoabelian group; this is weaker than solvable, which is the case
is finite (a natural number).
, any automorphism of G induces an automorphism of the abelianization. Since the abelianization is abelian, inner automorphisms act trivially, hence this yields a map
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, more specifically in abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, the commutator subgroup or derived subgroup of a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
is the subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called 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 x H is a group operation on H...
generated
Generating set of a group
In abstract algebra, a generating set of a group is a subset that is not contained in any proper subgroup of the group. Equivalently, a generating set of a group is a subset such that every element of the group can be expressed as the combination of finitely many elements of the subset and their...
by all the commutator
Commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:...
s of the group.
The commutator subgroup is important because it is the smallest normal subgroup
Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
such that the quotient group
Quotient group
In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...
of the original group by this subgroup is abelian
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
.
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 commutatorCommutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:...
of g and h is
. The commutator [g,h] is equal to the identity element e if and only if gh = hg, that is, if and only if g and h commute. In general, .An element of G which is of the form [g,h] for some g and h is called a commutator. The identity element
is always a commutator, and it is the only commutator if and only if G is abelianAbelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
.
Here are some simple but useful commutator identities, true for any elements s, g, h of a group G:
-
-
, where 
. - For any homomorphism
, 
The first and second identities imply that the set of commutators in G is closed under inversion and under conjugation. If in the third identity we take
, 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 .However, the product of two or more commutators need not be a commutator. A generic example is
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 [G,G] (also called the derived subgroup, and denoted G′ or G(1)) of G: it is the subgroup generatedGenerating set of a group
In abstract algebra, a generating set of a group is a subset that is not contained in any proper subgroup of the group. Equivalently, a generating set of a group is a subset such that every element of the group can be expressed as the combination of finitely many elements of the subset and their...
by all the commutators.
It follows from the properties of commutators that any element of [G,G] is of the form
for some natural number n. Moreover, since
, the commutator subgroup is normal in G. For any homomorphism ,,so that
.This shows that the commutator subgroup can be viewed as a functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
on the category of groups, some implications of which are explored below. Moreover, taking
it shows that the commutator subgroup is stable under every endomorphism of G: that is, [G,G] is a fully characteristic subgroupFully characteristic subgroup
In mathematics, a subgroup of a group is fully characteristic if it is invariant under every endomorphism of the group. That is, any endomorphism of the group takes elements of the subgroup to elements of the 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 = g1g2...gk that can be rearranged to give the identity.
Derived series
This construction can be iterated:The groups
are called the second derived subgroup, third derived subgroup, and so forth, and the descending normal seriesNormal series
In mathematics, a subgroup series is a chain of subgroups:1 = A_0 \leq A_1 \leq \cdots \leq A_n = G.Subgroup series can simplify the study of a group to the study of simpler subgroups and their relations, and several subgroup series can be invariantly defined and are important invariants of groups...
is called the derived series. This should not be confused with the lower central series, whose terms are
, not .For a finite group, the derived series terminates in a perfect group
Perfect group
In mathematics, more specifically in the area of modern algebra known as group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no nontrivial abelian quotients...
, 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 number
Ordinal number
In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
s via transfinite recursion, thereby obtaining the transfinite derived series, which eventually terminates at the perfect core
Perfect core
In mathematics, in the field of group theory, the perfect core of a group is its largest perfect subgroup. Its existence is guaranteed by the fact that the subgroup generated by a family of perfect subgroups is again a perfect subgroup...
of the group.
Abelianization
Given a group G, a factor group G/N is abelian if and only if [G,G] ≤ N.The quotient
is an abelian group called the abelianization of G or G made abelian. It is usually denoted by Gab or Gab.There is a useful categorical interpretation of the map
. Namely is universal for homomorphisms fromG to an abelian group H: for any abelian group H and homomorphism of groups
there exists a unique homomorphism such that . As usualfor objects defined by universal mapping properties, this shows the uniqueness of the
abelianization
up to canonical isomorphism, whereas theexplicit construction
shows existence.The abelianization functor is the left adjoint
Adjoint functors
In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another, called an adjunction. The relationship of adjunction is ubiquitous in mathematics, as it rigorously reflects the intuitive notions of optimization and efficiency...
of the inclusion functor from the category of abelian groups to the category of groups.
Another important interpretation of
is as, the first homology group of G with integral coefficients.Classes of groups
A group G is an abelian groupAbelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
if and only if the derived group is trivial:
. 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
Perfect group
In mathematics, more specifically in the area of modern algebra known as group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no nontrivial abelian quotients...
if and only if the derived group equals the group itself:
. Equivalently, if and only if the abelianization of the group is trivial. This is "opposite" to abelian.A group with
for some n in N is called a solvable groupSolvable group
In mathematics, more specifically in the field of group theory, a solvable group is a group that can be constructed from abelian groups using extensions...
; this is weaker than abelian, which is the case
.A group with
for some ordinal numberOrdinal number
In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
, possibly infinite, is called a hypoabelian group; this is weaker than solvable, which is the case
is finite (a natural number).Examples
- The commutator subgroup of the alternating group A4 is the Klein four group.
- The commutator subgroup of the symmetric groupSymmetric groupIn mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
is the alternating group 
. - The commutator subgroup of the quaternion groupQuaternion groupIn group theory, the quaternion group is a non-abelian group of order eight, isomorphic to a certain eight-element subset of the quaternions under multiplication...
Q = {1, −1, i, −i, j, −j, k, −k} is [Q,Q]={1, −1}.
Map from Out
Since the derived subgroup is characteristicCharacteristic subgroup
In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is invariant under all automorphisms of the parent group. Because conjugation is an automorphism, every characteristic subgroup is normal, though not every normal...
, any automorphism of G induces an automorphism of the abelianization. Since the abelianization is abelian, inner automorphisms act trivially, hence this yields a map