# 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.^{[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. 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.

## Contents

## Commutators

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For elements *g* and *h* of a group *G*, the commutator 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, *gh* = *hg*[*g*,*h*].

An element of *G* which is of the form [*g*,*h*] 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*:

- , where , the conjugate of g by s.
- For any homomorphism
*f*:*G*→*H*,*f*([*g*,*h*]) = [*f*(*g*),*f*(*h*)].

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 *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 .

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.^{[3]}

## 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 generated 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*, where the *g*_{i} and *h*_{i} are elements of *G*. Moreover, since , the commutator subgroup is normal in *G*. For any homomorphism *f*: *G* → *H*,

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* = *g*_{1} *g*_{2} ... *g _{k}* 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 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 numbers via transfinite recursion, thereby obtaining the **transfinite derived series**, which eventually terminates at the perfect core of the group.

### Abelianization

Given a group *G*, a quotient group *G*/*N* is abelian if and only if [*G*,*G*] ≤ *N*.

The quotient *G*/[*G*,*G*] is an abelian group called the **abelianization** of *G* or *G* **made abelian**.^{[4]} It is usually denoted by *G*^{ab} or *G*_{ab}.

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 *f*: *G* → *H* there exists a unique homomorphism *F*: *G*^{ab} → *H* such that . As usual for objects defined by universal mapping properties, this shows the uniqueness of the abelianization *G*^{ab} up to canonical isomorphism, whereas the explicit construction *G* → *G*/[*G*,*G*] 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 is as *H*_{1}(*G*,**Z**), the first homology group of *G* with integral coefficients.

### 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 for some *n* in **N** is called a **solvable group**; this is weaker than abelian, which is the case *n* = 1.

A group with for any *n* in **N** is called a **non solvable group**.

A group with 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).

## Examples

- The commutator subgroup of the alternating group
*A*_{4}is the Klein four group. - The commutator subgroup of the symmetric group
*S*is the alternating group_{n}*A*._{n} - The commutator subgroup of the quaternion group
*Q*= {1, −1,*i*, −*i*,*j*, −*j*,*k*, −*k*} is [*Q*,*Q*]={1, −1}. - The commutator subgroup of the fundamental group π
_{1}(*X*) of a path-connected topological space*X*is the kernel of the natural homomorphism onto the first singular homology group*H*_{1}(*X*).

### 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

## See also

- solvable group
- nilpotent group
- The abelianization H/H' of a subgroup H<G of finite index (G:H) is the target of the Artin transfer T(G,H)

## Notes

## References

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