# Commutator subgroup

__: Smallest normal subgroup by which the quotient is commutative__

**Short description**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, [math]\displaystyle{ G/N }[/math] is abelian if and only if [math]\displaystyle{ N }[/math] contains the commutator subgroup of [math]\displaystyle{ G }[/math]. 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 [math]\displaystyle{ g }[/math] and [math]\displaystyle{ h }[/math] of a group *G*, the commutator of [math]\displaystyle{ g }[/math] and [math]\displaystyle{ h }[/math] is [math]\displaystyle{ [g,h] = g^{-1}h^{-1}gh }[/math]. The commutator [math]\displaystyle{ [g,h] }[/math] is equal to the identity element *e* if and only if [math]\displaystyle{ gh = hg }[/math] , that is, if and only if [math]\displaystyle{ g }[/math] and [math]\displaystyle{ h }[/math] commute. In general, [math]\displaystyle{ gh = hg[g,h] }[/math].

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: [math]\displaystyle{ [g,h] = ghg^{-1}h^{-1} }[/math] in which case [math]\displaystyle{ gh \neq hg[g,h] }[/math] but instead [math]\displaystyle{ gh = [g,h]hg }[/math].

An element of *G* of the form [math]\displaystyle{ [g,h] }[/math] 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*:

- [math]\displaystyle{ [g,h]^{-1} = [h,g], }[/math]
- [math]\displaystyle{ [g,h]^s = [g^s,h^s], }[/math] where [math]\displaystyle{ g^s = s^{-1}gs }[/math] (or, respectively, [math]\displaystyle{ g^s = sgs^{-1} }[/math]) is the conjugate of [math]\displaystyle{ g }[/math] by [math]\displaystyle{ s, }[/math]
- for any homomorphism [math]\displaystyle{ f: G \to H }[/math], [math]\displaystyle{ f([g, h]) = [f(g), f(h)]. }[/math]

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*, [math]\displaystyle{ x \mapsto x^s }[/math], 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.^{[3]}

## Definition

This motivates the definition of the **commutator subgroup** [math]\displaystyle{ [G, G] }[/math] (also called the **derived subgroup**, and denoted [math]\displaystyle{ G' }[/math] or [math]\displaystyle{ G^{(1)} }[/math]) of *G*: it is the subgroup generated by all the commutators.

It follows from this definition that any element of [math]\displaystyle{ [G, G] }[/math] is of the form

- [math]\displaystyle{ [g_1,h_1] \cdots [g_n,h_n] }[/math]

for some natural number [math]\displaystyle{ n }[/math], where the *g*_{i} and *h*_{i} are elements of *G*. Moreover, since [math]\displaystyle{ ([g_1,h_1] \cdots [g_n,h_n])^s = [g_1^s,h_1^s] \cdots [g_n^s,h_n^s] }[/math], the commutator subgroup is normal in *G*. For any homomorphism *f*: *G* → *H*,

- [math]\displaystyle{ f([g_1,h_1] \cdots [g_n,h_n]) = [f(g_1),f(h_1)] \cdots [f(g_n),f(h_n)] }[/math],

so that [math]\displaystyle{ f([G,G]) \subseteq [H,H] }[/math].

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:

- [math]\displaystyle{ G^{(0)} := G }[/math]
- [math]\displaystyle{ G^{(n)} := [G^{(n-1)},G^{(n-1)}] \quad n \in \mathbf{N} }[/math]

The groups [math]\displaystyle{ G^{(2)}, G^{(3)}, \ldots }[/math] are called the **second derived subgroup**, **third derived subgroup**, and so forth, and the descending normal series

- [math]\displaystyle{ \cdots \triangleleft G^{(2)} \triangleleft G^{(1)} \triangleleft G^{(0)} = G }[/math]

is called the **derived series**. This should not be confused with the **lower central series**, whose terms are [math]\displaystyle{ G_n := [G_{n-1},G] }[/math].

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 [math]\displaystyle{ G }[/math], a quotient group [math]\displaystyle{ G/N }[/math] is abelian if and only if [math]\displaystyle{ [G, G]\subseteq N }[/math].

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

There is a useful categorical interpretation of the map [math]\displaystyle{ \varphi: G \rightarrow G^{\operatorname{ab}} }[/math]. Namely [math]\displaystyle{ \varphi }[/math] is universal for homomorphisms from [math]\displaystyle{ G }[/math] to an abelian group [math]\displaystyle{ H }[/math]: for any abelian group [math]\displaystyle{ H }[/math] and homomorphism of groups [math]\displaystyle{ f: G \to H }[/math] there exists a unique homomorphism [math]\displaystyle{ F: G^{\operatorname{ab}}\to H }[/math] such that [math]\displaystyle{ f = F \circ \varphi }[/math]. As usual for objects defined by universal mapping properties, this shows the uniqueness of the abelianization [math]\displaystyle{ G^{\operatorname{ab}} }[/math] up to canonical isomorphism, whereas the explicit construction [math]\displaystyle{ G\to G/[G, G] }[/math] 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 [math]\displaystyle{ G^{\operatorname{ab}} }[/math] is as [math]\displaystyle{ H_1(G, \mathbb{Z}) }[/math], the first homology group of [math]\displaystyle{ G }[/math] with integral coefficients.

### Classes of groups

A group [math]\displaystyle{ G }[/math] 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 [math]\displaystyle{ G }[/math] 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 [math]\displaystyle{ G^{(n)}=\{e\} }[/math] for some *n* in **N** is called a **solvable group**; this is weaker than abelian, which is the case *n* = 1.

A group with [math]\displaystyle{ G^{(n)} \neq \{e\} }[/math] for all *n* in **N** is called a **non-solvable group**.

A group with [math]\displaystyle{ G^{(\alpha)}=\{e\} }[/math] 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 [math]\displaystyle{ G }[/math] has derived subgroup equal to itself, [math]\displaystyle{ G^{(1)} =G }[/math], it is called a **perfect group**. This includes non-abelian simple groups and the special linear groups [math]\displaystyle{ \operatorname{SL}_n(k) }[/math] for a fixed field [math]\displaystyle{ k }[/math].

## Examples

- The commutator subgroup of any abelian group is trivial.
- The commutator subgroup of the general linear group [math]\displaystyle{ \operatorname{GL}_n(k) }[/math] over a field or a division ring
*k*equals the special linear group [math]\displaystyle{ \operatorname{SL}_n(k) }[/math] provided that [math]\displaystyle{ n \ne 2 }[/math] or*k*is not the field with two elements.^{[5]} - 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}.

### 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

- [math]\displaystyle{ \operatorname{Out}(G) \to \operatorname{Aut}(G^{\mbox{ab}}) }[/math]

## 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

- Dummit, David S.; Foote, Richard M. (2004),
*Abstract Algebra*(3rd ed.),*John Wiley & Sons*, ISBN 0-471-43334-9 - Fraleigh, John B. (1976),
*A First Course In Abstract Algebra*(2nd ed.), Reading: Addison-Wesley, ISBN 0-201-01984-1 - Lang, Serge (2002),
*Algebra*, Graduate Texts in Mathematics, Springer, ISBN 0-387-95385-X - Suárez-Alvarez, Mariano. "Derived Subgroups and Commutators". https://math.stackexchange.com/q/7811.

## External links

- Hazewinkel, Michiel, ed. (2001), "Commutator subgroup",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/c023440

Original source: https://en.wikipedia.org/wiki/Commutator subgroup.
Read more |