Commutator subgroup

2023-04-24T17:10:20Z

192.76.8.67: /* Examples */

{{short description|Smallest normal subgroup by which the quotient is commutative}}
In [[mathematics]], more specifically in [[abstract algebra]], the '''commutator subgroup''' or '''derived subgroup''' of a [[group (mathematics)|group]] is the [[subgroup (mathematics)|subgroup]] [[generating set of a group|generated]] by all the [[commutator]]s of the group.<ref>{{harvtxt|Dummit|Foote|2004}}</ref><ref>{{harvtxt|Lang|2002}}</ref>

The commutator subgroup is important because it is the [[Universal property|smallest]] [[normal subgroup]] such that the [[quotient group]] of the original group by this subgroup is [[abelian group|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.

== Commutators ==
{{main|Commutator}}
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 [[Conjugacy class|conjugate]] of <math>g</math> by <math>s,</math>
* for any [[Group homomorphism|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 (mathematics)|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.<ref>{{harvtxt|Suárez-Alvarez}}</ref>

== Definition ==
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 [[generating set of a group|generated]] by all the commutators.

It follows from this definition 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 <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]) \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>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 number]]s via [[transfinite recursion]], thereby obtaining the '''transfinite derived series''', which eventually terminates at the [[perfect core]] of the group.

=== Abelianization ===
Given a group <math>G</math>, a [[quotient group]] <math>G/N</math> is abelian if and only if <math>[G, G]\subseteq 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'''.<ref>{{harvtxt|Fraleigh|1976|p=108}}</ref> 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 [[adjoint functors|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 [[group homology|homology group]] of <math>G</math> with integral coefficients.

=== Classes of groups ===
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 '''[[perfect radical|hypoabelian group]]'''; this is weaker than solvable, which is the case ''α'' is finite (a natural number).

=== Perfect group ===
{{Main articles|Perfect group}}
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 group|simple groups]] and the [[Special linear group|special linear groups]] <math>\operatorname{SL}_n(k)</math> for a fixed field <math>k</math>.

== Examples ==
* The commutator subgroup of any [[abelian group]] is [[Trivial group|trivial]].
* The commutator subgroup of the [[general linear group]] <math>\operatorname{GL}_n(k)</math> over a [[Field (mathematics)|field]] or a [[division ring]] ''k'' equals the [[special linear group]] <math>\operatorname{SL}_n(k)</math> provided that <math>n \ne 2</math> or ''k'' is not the [[finite field|field with two elements]].<ref>{{citation|author=Suprunenko|first=D.A.|title=Matrix groups|publisher=American Mathematical Society|year=1976|series=Translations of Mathematical Monographs}}, Theorem II.9.4</ref>
* The commutator subgroup of the [[alternating group]] ''A''4 is the [[Klein four group]].
* The commutator subgroup of the [[symmetric group]] ''Sn'' is the [[alternating group]] ''An''.
* 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 subgroup|characteristic]], any automorphism of ''G'' induces an automorphism of the abelianization. Since the abelianization is abelian, [[inner automorphism]]s act trivially, hence this yields a map
:<math>\operatorname{Out}(G) \to \operatorname{Aut}(G^{\mbox{ab}})</math>

==See also==
*[[Solvable group]]
*[[Nilpotent group]]
*The abelianization ''H''/''H''<nowiki>'</nowiki> of a subgroup ''H'' < ''G'' of finite [[Index of a subgroup|index]] (''G'':''H'') is the [[Artin transfer (group theory)#Artin transfer|target of the Artin transfer]] ''T''(''G'',''H'').

== Notes ==
<references/>

== References ==
* {{ citation | last1 = Dummit | first1 = David S. | last2 = Foote | first2 = Richard M. | title = Abstract Algebra | publisher = [[John Wiley & Sons]] | year = 2004 | edition = 3rd | isbn = 0-471-43334-9 }}
* {{ citation | first1 = John B. | last1 = Fraleigh | year = 1976 | isbn = 0-201-01984-1 | title = A First Course In Abstract Algebra | edition = 2nd | publisher = [[Addison-Wesley]] | location = Reading }}
* {{citation | last = Lang | first = Serge | author-link = Serge Lang | title = Algebra | publisher = [[Springer Science+Business Media|Springer]] | series = [[Graduate Texts in Mathematics]] | year = 2002 | isbn = 0-387-95385-X}}
* {{ cite web | url = https://math.stackexchange.com/q/7811 | first = Mariano | last = Suárez-Alvarez | title = Derived Subgroups and Commutators }}

==External links==
* {{springer|title=Commutator subgroup|id=p/c023440}}

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Commutator subgroup