Inmathematics, more specifically inabstract algebra, thecommutator subgroup orderived subgroup of agroup is thesubgroupgenerated by all thecommutators of the group.^[1][2]

The commutator subgroup is important because it is thesmallestnormal subgroup such that thequotient group of the original group by this subgroup isabelian. In other words,${\displaystyle G/N}$ is abelianif and only if${\displaystyle N}$ contains the commutator subgroup of${\displaystyle G}$. 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.

For elements${\displaystyle g}$ and${\displaystyle h}$ of a groupG, thecommutator of${\displaystyle g}$ and${\displaystyle h}$ is${\displaystyle [g,h]=g^{-1}{h}^{-1}gh}$. The commutator${\displaystyle [g,h]}$ is equal to theidentity elemente if and only if${\displaystyle gh=hg}$ , that is, if and only if${\displaystyle g}$ and${\displaystyle h}$ commute. In general,${\displaystyle gh=hg[g,h]}$.

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:${\displaystyle [g,h]=gh{g}^{-1}{h}^{-1}}$ in which case${\displaystyle gh\ne hg[g,h]}$ but instead${\displaystyle gh=[g,h]hg}$.

An element ofG of the form${\displaystyle [g,h]}$ for someg andh is called a commutator. The identity elemente = [e,e] is always a commutator, and it is the only commutator if and only ifG is abelian.

Here are some simple but useful commutator identities, true for any elementss,g,h of a groupG:

• ${\displaystyle [g,h{]}^{-1}=[h,g],}$
• ${\displaystyle [g,h{]}^s=[g^s,h^s],}$ where${\displaystyle g^s=s^{-1}gs}$ (or, respectively,${\displaystyle g^s=sg{s}^{-1}}$) is theconjugate of${\displaystyle g}$ by${\displaystyle s,}$
• for anyhomomorphism${\displaystyle f:G\to H}$,${\displaystyle f([g,h])=[f(g),f(h)].}$

The first and second identities imply that theset of commutators inG is closed under inversion and conjugation. If in the third identity we takeH =G, we get that the set of commutators is stable under anyendomorphism ofG. This is in fact a generalization of the second identity, since we can takef to be the conjugationautomorphism onG,${\displaystyle x\mapsto x^s}$, 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 thefree group ona,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]

This motivates the definition of thecommutator subgroup${\displaystyle [G,G]}$ (also called thederived subgroup, and denoted${\displaystyle G^{\prime }}$ or${\displaystyle G^{(1)}}$) ofG: it is the subgroupgenerated by all the commutators.

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

${\displaystyle [g_1,h_1]\cdots [g_n,h_n]}$

for somenatural number${\displaystyle n}$, where theg_i andh_i are elements ofG. Moreover, since${\displaystyle ([g_1,h_1]\cdots [g_n,h_n]{)}^s=[g_1^s,h_1^s]\cdots [g_n^s,h_n^s]}$, the commutator subgroup is normal inG. For any homomorphismf:G →H,

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

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

This shows that the commutator subgroup can be viewed as afunctor on thecategory of groups, some implications of which are explored below. Moreover, takingG =H it shows that the commutator subgroup is stable under every endomorphism ofG: that is, [G,G] is afully characteristic subgroup ofG, a property considerably stronger than normality.

The commutator subgroup can also be defined as the set of elementsg of the group that have an expression as a productg =g₁g₂ ...g_k that can be rearranged to give the identity.

This construction can be iterated:

${\displaystyle G^{(0)}:=G}$

${\displaystyle G^{(n)}:=[G^{(n-1)},G^{(n-1)}]n\in \mathbf{N}}$

The groups${\displaystyle G^{(2)},G^{(3)},\dots }$ are called thesecond derived subgroup,third derived subgroup, and so forth, and the descendingnormal series

${\displaystyle \cdots ◃G^{(2)}◃G^{(1)}◃G^{(0)}=G}$

is called thederived series. This should not be confused with thelower central series, whose terms are${\displaystyle G_n:=[G_{n-1},G]}$.

For a finite group, the derived series terminates in aperfect 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 infiniteordinal numbers viatransfinite recursion, thereby obtaining thetransfinite derived series, which eventually terminates at theperfect core of the group.

Given a group${\displaystyle G}$, aquotient group${\displaystyle G/N}$ is abelian if and only if${\displaystyle [G,G]\subseteq N}$.

The quotient${\displaystyle G/[G,G]}$ is an abelian group called theabelianization of${\displaystyle G}$ or${\displaystyle G}$made abelian.^[4] It is usually denoted by${\displaystyle G^{\mathrm{ab}}}$ or${\displaystyle G_{\mathrm{ab}}}$.

There is a useful categorical interpretation of the map${\displaystyle \phi :G\to G^{\mathrm{ab}}}$. Namely${\displaystyle \phi }$ is universal for homomorphisms from${\displaystyle G}$ to an abelian group${\displaystyle H}$: for any abelian group${\displaystyle H}$ and homomorphism of groups${\displaystyle f:G\to H}$ there exists a unique homomorphism${\displaystyle F:G^{\mathrm{ab}}\to H}$ such that${\displaystyle f=F\circ \phi }$. As usual for objects defined by universal mapping properties, this shows the uniqueness of the abelianization${\displaystyle G^{\mathrm{ab}}}$ up to canonical isomorphism, whereas the explicit construction${\displaystyle G\to G/[G,G]}$ shows existence.

The abelianization functor is theleft adjoint of the inclusion functor from thecategory of abelian groups to the category of groups. The existence of the abelianization functorGrp →Ab makes the categoryAb areflective subcategory of the category of groups, defined as a full subcategory whose inclusion functor has a left adjoint.

Another important interpretation of${\displaystyle G^{\mathrm{ab}}}$ is as${\displaystyle H_1(G,\mathbb{Z})}$, the firsthomology group of${\displaystyle G}$ with integral coefficients.

A group${\displaystyle G}$ is anabelian 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${\displaystyle G}$ is aperfect 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${\displaystyle G^{(n)}=\{e\}}$ for somen inN is called asolvable group; this is weaker than abelian, which is the casen = 1.

A group with${\displaystyle G^{(n)}\ne \{e\}}$ for alln inN is called anon-solvable group.

A group with${\displaystyle G^{(\alpha )}=\{e\}}$ for someordinal number, possibly infinite, is called ahypoabelian group; this is weaker than solvable, which is the caseα is finite (a natural number).

Whenever a group${\displaystyle G}$ has derived subgroup equal to itself,${\displaystyle G^{(1)}=G}$, it is called aperfect group. This includes non-abeliansimple groups and thespecial linear groups${\displaystyle {\mathrm{SL}}_n(k)}$ for a fixed field${\displaystyle k}$.

• The commutator subgroup of anyabelian group istrivial.
• The commutator subgroup of thegeneral linear group${\displaystyle {\mathrm{GL}}_n(k)}$ over afield or adivision ringk equals thespecial linear group${\displaystyle {\mathrm{SL}}_n(k)}$ provided that${\displaystyle n\ne 2}$ ork is not thefield with two elements.^[5]
• The commutator subgroup of thealternating groupA₄ is theKlein four group.
• The commutator subgroup of thesymmetric groupS_n is thealternating groupA_n.
• The commutator subgroup of thequaternion groupQ = {1, −1,i, −i,j, −j,k, −k} is [Q,Q] = {1, −1}.

Since the derived subgroup ischaracteristic, any automorphism ofG induces an automorphism of the abelianization. Since the abelianization is abelian,inner automorphisms act trivially, hence this yields a map

${\displaystyle \mathrm{Out}(G)\to \mathrm{Aut}(G^{\text{ab}})}$

• Solvable group
• Nilpotent group
• The abelianizationH/H' of a subgroupH < G of finiteindex (G:H) is thetarget of the Artin transfer T(G,H).

• 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".

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

• Group theory
• Functional subgroups
• Subgroup properties

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