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>G/N</math> is abelian if and only if <math>N</math> contains the commutator subgroup of <math>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.

## CommutatorsEdit

For elements <math>g</math> and <math>h</math> of a group *G*, the commutator of <math>g</math> and <math>h</math> is <math>[g,h] = g^{-1}h^{-1}gh</math>. The commutator <math>[g,h]</math> is equal to the identity element *e* if and only if <math>gh = hg</math> , that is, if and only if <math>g</math> and <math>h</math> commute. In general, <math>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>[g,h] = ghg^{-1}h^{-1}</math> in which case <math>gh \neq hg[g,h]</math> but instead <math>gh = [g,h]hg</math>.

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

## DefinitionEdit

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

It follows from the properties of commutators that any element of <math>[G, G]</math> is of the form

- <math>[g_1,h_1] \cdots [g_n,h_n] </math>

for some natural number <math>n</math>, where the *g*_{i} and *h*_{i} are elements of *G*. Moreover, since for any *s* in *G* we have <math>([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>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>f([G,G]) \leq [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 seriesEdit

This construction can be iterated:

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

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

- <math>\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>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.

### AbelianizationEdit

Given a group <math>G</math>, a quotient group <math>G/N</math> is abelian if and only if <math>[G, G]\leq N</math>.

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

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

### Classes of groupsEdit

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

A group with <math>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 groupEdit

Whenever a group <math>G</math> has derived subgroup equal to itself, <math>G^{(1)} =G</math>, it is called a **perfect group**. This includes non-abelian simple groups and the special linear groups <math>\text{SL}_n(k)</math> for a fixed field <math>k</math>.

## ExamplesEdit

- The commutator subgroup of any abelian group is trivial.
- The commutator subgroup of the general linear group <math>GL_n(k)</math> over a field or a division ring
*k*equals the special linear group <math>SL_n(k)</math> provided that <math>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}. - 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 OutEdit

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>\mbox{Out}(G) \to \mbox{Aut}(G^{\mbox{ab}})</math>

## See alsoEdit

- 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*).

## NotesEdit

- ↑ Dummit & Foote (2004)
- ↑ Lang (2002)
- ↑ Suárez-Alvarez
- ↑ Fraleigh (1976, p. 108)
- ↑ Suprunenko, D.A. (1976),
*Matrix groups*, Translations of Mathematical Monographs, American Mathematical Society, Theorem II.9.4

## ReferencesEdit

- 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".CS1 maint: ref=harv (link)

## External linksEdit

- "Commutator subgroup",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994]