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''of a group, derived group, second term of the lower central series, of a group''
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{{TEX|done}}
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{{MSC|20}}
  
The subgroup of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c0234401.png" /> generated by all commutators of the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c0234402.png" /> (cf. [[Commutator|Commutator]]). The commutator subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c0234403.png" /> is usually denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c0234404.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c0234405.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c0234406.png" />. The commutator subgroup is a [[Fully-characteristic subgroup|fully-characteristic subgroup]], and any subgroup containing the commutator subgroup is a normal subgroup. The quotient group with respect to some normal subgroup is Abelian if and only if this normal subgroup contains the commutator subgroup of the group.
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The commutator subgroup of a group $G$,
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also called the derived group, second term of the lower central series, of  $G$, is
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the subgroup of  $G$ generated by all commutators of the elements of $G$ (cf.
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[[Commutator|Commutator]]). The commutator subgroup of $G$ is usually denoted by $[G,G]$, $G'$ or $\Gamma_2(G)$. The commutator subgroup is a
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[[Fully-characteristic subgroup|fully-characteristic subgroup]], and any subgroup containing the commutator subgroup is a normal subgroup. The quotient group with respect to some normal subgroup is Abelian if and only if this normal subgroup contains the commutator subgroup of the group.
  
The commutator ideal of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c0234407.png" /> is the ideal generated by all products <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c0234408.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c0234409.png" />; it is also called the square of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c02344010.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c02344011.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c02344012.png" />.
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The commutator ideal of a ring $R$ is the ideal generated by all products $ab$, $a,b \in R$; it is also called the square of $R$ and is denoted by $[R,R]$ or $R^2$.
 
 
Both the above concepts are special cases of the notion of the commutator subgroup of a multi-operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c02344014.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c02344015.png" />, which is defined as the ideal generated by all commutators and all elements of the form
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c02344016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
 
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c02344017.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c02344018.png" />-ary operation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c02344019.png" /> and
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c02344020.png" /></td> </tr></table>
 
  
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Both the above concepts are special cases of the notion of the commutator subgroup of a multi-operator $\def\O{\Omega}\O$-group $G$, which is defined as the ideal generated by all commutators and all elements of the form
  
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$$\def\o{\omega} -a_1\cdots a_n\o - b_1\cdots b_n\o + (a_1+b_2)\cdots(a_n+b_n)\o, \tag{$*$}$$
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where $\o$ is an $n$-ary operation in $\O$ and
  
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$$a_,\dots,a_n,b_1,\dots,b_n\in G.$$
 
====Comments====
 
====Comments====
In the case of a ring considered as an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c02344021.png" />-group the commutators (of the underlying commutative group) are all zero, so that the commutator ideal is the ideal generated by all elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c02344022.png" />. Hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c02344023.png" /> is the ideal generated by all products <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c02344024.png" />.
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In the case of a ring considered as an operator $\O$-group the commutators (of the underlying commutative group) are all zero, so that the commutator ideal is the ideal generated by all elements $-a_1a_2 - b_1b_2 + (a_1+a_2)(b_1+b_2) = a_1b_2 + a_2b_1$. Hence $[R,R]$ is the ideal generated by all products $ab$.
  
More generally, in all three cases one defines the commutator group (ideal) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c02344025.png" /> of two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c02344026.png" />-subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c02344027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c02344028.png" /> as the ideal generated by all commutators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c02344029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c02344030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c02344031.png" />, and all elements (*) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c02344032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c02344033.png" />.
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More generally, in all three cases one defines the commutator group (ideal) $[A,B]$ of two $\O$-subgroups $A$ and $B$ as the ideal generated by all commutators $[a,b]$, $a\in A$, $b\in B$, and all elements (*) with $a_1,\dots,a_n\in A$, $b_1,\dots,b_n\in B$.
  
In the case of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c02344034.png" /> there is a second, different notion which also goes by the name of commutator ideal. It is the ideal generated by all commutators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c02344035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c02344036.png" />. This one is universal for homomorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c02344037.png" /> into commutative rings. I.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c02344038.png" /> is this ideal and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c02344039.png" /> is the natural projection, then for each homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c02344040.png" /> into a commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c02344041.png" /> there is a unique homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c02344042.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c02344043.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c02344044.png" /> factors uniquely through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c02344045.png" />). This is analogous to the property that for ordinary groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c02344046.png" /> is universal for mappings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023440/c02344047.png" /> into Abelian groups (cf. [[Universal problems|Universal problems]]).
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In the case of a ring $R$ there is a second, different notion which also goes by the name of commutator ideal. It is the ideal generated by all commutators $ab-ba$, $a,b\in R$. This one is universal for homomorphisms of $R$ into commutative rings. I.e. if $\mathfrak{c}$ is this ideal and $\pi : R \to R^{\textrm{ab}} = R/\mathfrak{c}$ is the natural projection, then for each homomorphism $g:R\to A$ into a commutative ring $A$ there is a unique homomorphism $g':R^{\textrm{ab}} \to A$ such that $g = g'\circ \pi$ ($g$ factors uniquely through $\pi$). This is analogous to the property that for ordinary groups $G\to G^{\textrm{ab}} = G/[G,G]$ is universal for mappings of $G$ into Abelian groups (cf.
 +
[[Universal problems|Universal problems]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P.M. Cohn,  "Algebra" , '''2''' , Wiley  (1977)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A.G. Kurosh,  "Lectures on general algebra" , Chelsea  (1963)  (Translated from Russian)</TD></TR></table>
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{|
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|-
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|valign="top"|{{Ref|Co}}||valign="top"| P.M. Cohn,  "Algebra", '''2''', Wiley  (1977) {{MR|0530404}}  {{ZBL|0341.00002}}
 +
|-
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|valign="top"|{{Ref|Ku}}||valign="top"| A.G. Kurosh,  "Lectures on general algebra", Chelsea  (1963)  (Translated from Russian) {{MR|0158000}} 
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|-
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|}

Revision as of 10:27, 1 March 2012

2020 Mathematics Subject Classification: Primary: 20-XX [MSN][ZBL]

The commutator subgroup of a group $G$, also called the derived group, second term of the lower central series, of $G$, is the subgroup of $G$ generated by all commutators of the elements of $G$ (cf. Commutator). The commutator subgroup of $G$ is usually denoted by $[G,G]$, $G'$ or $\Gamma_2(G)$. The commutator subgroup is a fully-characteristic subgroup, and any subgroup containing the commutator subgroup is a normal subgroup. The quotient group with respect to some normal subgroup is Abelian if and only if this normal subgroup contains the commutator subgroup of the group.

The commutator ideal of a ring $R$ is the ideal generated by all products $ab$, $a,b \in R$; it is also called the square of $R$ and is denoted by $[R,R]$ or $R^2$.

Both the above concepts are special cases of the notion of the commutator subgroup of a multi-operator $\def\O{\Omega}\O$-group $G$, which is defined as the ideal generated by all commutators and all elements of the form

$$\def\o{\omega} -a_1\cdots a_n\o - b_1\cdots b_n\o + (a_1+b_2)\cdots(a_n+b_n)\o, \tag{$*$}$$ where $\o$ is an $n$-ary operation in $\O$ and

$$a_,\dots,a_n,b_1,\dots,b_n\in G.$$

Comments

In the case of a ring considered as an operator $\O$-group the commutators (of the underlying commutative group) are all zero, so that the commutator ideal is the ideal generated by all elements $-a_1a_2 - b_1b_2 + (a_1+a_2)(b_1+b_2) = a_1b_2 + a_2b_1$. Hence $[R,R]$ is the ideal generated by all products $ab$.

More generally, in all three cases one defines the commutator group (ideal) $[A,B]$ of two $\O$-subgroups $A$ and $B$ as the ideal generated by all commutators $[a,b]$, $a\in A$, $b\in B$, and all elements (*) with $a_1,\dots,a_n\in A$, $b_1,\dots,b_n\in B$.

In the case of a ring $R$ there is a second, different notion which also goes by the name of commutator ideal. It is the ideal generated by all commutators $ab-ba$, $a,b\in R$. This one is universal for homomorphisms of $R$ into commutative rings. I.e. if $\mathfrak{c}$ is this ideal and $\pi : R \to R^{\textrm{ab}} = R/\mathfrak{c}$ is the natural projection, then for each homomorphism $g:R\to A$ into a commutative ring $A$ there is a unique homomorphism $g':R^{\textrm{ab}} \to A$ such that $g = g'\circ \pi$ ($g$ factors uniquely through $\pi$). This is analogous to the property that for ordinary groups $G\to G^{\textrm{ab}} = G/[G,G]$ is universal for mappings of $G$ into Abelian groups (cf. Universal problems).

References

[Co] P.M. Cohn, "Algebra", 2, Wiley (1977) MR0530404 Zbl 0341.00002
[Ku] A.G. Kurosh, "Lectures on general algebra", Chelsea (1963) (Translated from Russian) MR0158000
How to Cite This Entry:
Commutator subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Commutator_subgroup&oldid=21388
This article was adapted from an original article by N.N. Vil'yamsO.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article