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The subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096580/v0965801.png" /> of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096580/v0965802.png" /> generated by all possible values of all words (cf. [[Word|Word]]) of some set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096580/v0965803.png" />, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096580/v0965804.png" /> run through the entire group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096580/v0965805.png" /> independently of each other. A verbal subgroup is normal; the congruence defined on the group by a verbal subgroup is a [[Verbal congruence|verbal congruence]] (see also [[Algebraic systems, variety of|Algebraic systems, variety of]]).
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Examples of verbal subgroups: 1) the commutator subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096580/v0965806.png" /> of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096580/v0965807.png" /> defined by the word <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096580/v0965808.png" />; 2) the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096580/v0965809.png" />-th commutator subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096580/v09658010.png" />; 3) the terms of the lower central series
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<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/v/v096/v096580/v09658011.png" /></td> </tr></table>
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The subgroup  $  V( G) $
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of a group  $  G $
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generated by all possible values of all words (cf. [[Word|Word]]) of some set  $  V = \{ {f _  \nu  ( x _ {1} \dots x _ {n _  \nu  } ) } : {\nu \in I } \} $,
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when  $  x _ {1} , x _ {2} \dots $
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run through the entire group  $  G $
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independently of each other. A verbal subgroup is normal; the congruence defined on the group by a verbal subgroup is a [[Verbal congruence|verbal congruence]] (see also [[Algebraic systems, variety of|Algebraic systems, variety of]]).
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096580/v09658012.png" /> is the verbal subgroup defined by the commutator
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Examples of verbal subgroups: 1) the commutator subgroup $  G  ^  \prime  $
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of a group  $  G $
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defined by the word  $  [ x, y] = x  ^ {-1} y  ^ {-1} xy $;
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2) the  $  n $-th commutator subgroup  $  G  ^ {( n)} = {( G  ^ {( n- 1) } ) }  ^  \prime  $;
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3) the terms of the lower central series
  
<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/v/v096/v096580/v09658013.png" /></td> </tr></table>
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$$
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\Gamma _ {1} ( G)  = G  \supseteq  \Gamma _ {2} ( G)  \supseteq \dots
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\supseteq  \Gamma _ {n} ( G)  \supseteq \dots ,
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$$
  
4) the power subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096580/v09658014.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096580/v09658015.png" /> defined by the words <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096580/v09658016.png" />.
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where  $  \Gamma _ {n} ( G) $
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is the verbal subgroup defined by the commutator
  
The equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096580/v09658017.png" /> is valid for any homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096580/v09658018.png" />. In particular, every verbal subgroup is a [[Fully-characteristic subgroup|fully-characteristic subgroup]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096580/v09658019.png" />. The converse is true for free groups, but not in general: The intersection of two verbal subgroups may not be a verbal subgroup.
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$$
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[ x _ {1} \dots x _ {n} ]  = \
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[[ x _ {1} \dots x _ {n-1} ], x _ {n} ] ;
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$$
  
Verbal subgroups of the free group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096580/v09658020.png" /> of countable rank are especially important. They constitute a (modular) sublattice of the lattice of all its subgroups. Verbal subgroups are  "monotone" : If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096580/v09658021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096580/v09658022.png" /> (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096580/v09658023.png" /> means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096580/v09658024.png" /> is a normal subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096580/v09658025.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096580/v09658026.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096580/v09658027.png" />.
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4) the power subgroup  $  G  ^ {n} $
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of the group $  G $
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defined by the words  $  x  ^ {n} $.
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The equality  $  V( G) \phi = V( G \phi ) $
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is valid for any homomorphism  $  \phi $.  
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In particular, every verbal subgroup is a [[Fully-characteristic subgroup|fully-characteristic subgroup]] in  $  G $.  
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The converse is true for free groups, but not in general: The intersection of two verbal subgroups may not be a verbal subgroup.
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Verbal subgroups of the free group  $  X $
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of countable rank are especially important. They constitute a (modular) sublattice of the lattice of all its subgroups. Verbal subgroups are  "monotone" : If $  R \lhd X $,  
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$  S \lhd X $(
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here $  R \lhd X $
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means that $  R $
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is a normal subgroup of $  X $)  
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and $  R \subset  S $,  
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then $  V( R) \subset  V( S) $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,  "The theory of groups" , '''1–2''' , Chelsea  (1955–1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Neumann,  "Varieties of groups" , Springer  (1967)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,  "The theory of groups" , '''1–2''' , Chelsea  (1955–1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Neumann,  "Varieties of groups" , Springer  (1967)</TD></TR></table>

Latest revision as of 11:34, 12 January 2021


The subgroup $ V( G) $ of a group $ G $ generated by all possible values of all words (cf. Word) of some set $ V = \{ {f _ \nu ( x _ {1} \dots x _ {n _ \nu } ) } : {\nu \in I } \} $, when $ x _ {1} , x _ {2} \dots $ run through the entire group $ G $ independently of each other. A verbal subgroup is normal; the congruence defined on the group by a verbal subgroup is a verbal congruence (see also Algebraic systems, variety of).

Examples of verbal subgroups: 1) the commutator subgroup $ G ^ \prime $ of a group $ G $ defined by the word $ [ x, y] = x ^ {-1} y ^ {-1} xy $; 2) the $ n $-th commutator subgroup $ G ^ {( n)} = {( G ^ {( n- 1) } ) } ^ \prime $; 3) the terms of the lower central series

$$ \Gamma _ {1} ( G) = G \supseteq \Gamma _ {2} ( G) \supseteq \dots \supseteq \Gamma _ {n} ( G) \supseteq \dots , $$

where $ \Gamma _ {n} ( G) $ is the verbal subgroup defined by the commutator

$$ [ x _ {1} \dots x _ {n} ] = \ [[ x _ {1} \dots x _ {n-1} ], x _ {n} ] ; $$

4) the power subgroup $ G ^ {n} $ of the group $ G $ defined by the words $ x ^ {n} $.

The equality $ V( G) \phi = V( G \phi ) $ is valid for any homomorphism $ \phi $. In particular, every verbal subgroup is a fully-characteristic subgroup in $ G $. The converse is true for free groups, but not in general: The intersection of two verbal subgroups may not be a verbal subgroup.

Verbal subgroups of the free group $ X $ of countable rank are especially important. They constitute a (modular) sublattice of the lattice of all its subgroups. Verbal subgroups are "monotone" : If $ R \lhd X $, $ S \lhd X $( here $ R \lhd X $ means that $ R $ is a normal subgroup of $ X $) and $ R \subset S $, then $ V( R) \subset V( S) $.

References

[1] A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian)
[2] H. Neumann, "Varieties of groups" , Springer (1967)
How to Cite This Entry:
Verbal subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Verbal_subgroup&oldid=17324
This article was adapted from an original article by O.N. Golovin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article