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Difference between revisions of "Verbal subgroup"

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Examples of verbal subgroups: 1) the commutator subgroup  $  G  ^  \prime  $
 
Examples of verbal subgroups: 1) the commutator subgroup  $  G  ^  \prime  $
 
of a group  $  G $
 
of a group  $  G $
defined by the word  $  [ x, y] = x  ^ {-} 1 y  ^ {-} 1 xy $;  
+
defined by the word  $  [ x, y] = x  ^ {-1} y  ^ {-1} xy $;  
2) the  $  n $-
+
2) the  $  n $-th commutator subgroup  $  G  ^ {( n)} = {( G  ^ {( n- 1) } ) }  ^  \prime  $;  
th commutator subgroup  $  G  ^ {(} n) = {( G  ^ {(} n- 1) ) }  ^  \prime  $;  
 
 
3) the terms of the lower central series
 
3) the terms of the lower central series
  
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$$  
 
$$  
 
[ x _ {1} \dots x _ {n} ]  = \  
 
[ x _ {1} \dots x _ {n} ]  = \  
[[ x _ {1} \dots x _ {n-} 1 ], x _ {n} ] ;
+
[[ x _ {1} \dots x _ {n-1} ], x _ {n} ] ;
 
$$
 
$$
  
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Verbal subgroups of the free group  $  X $
 
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 riangle\left X \right .$,  
+
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 riangle\left X \right .$(
+
$  S \lhd X $(
here  $  R riangle\left X \right .$
+
here  $  R \lhd X $
 
means that  $  R $
 
means that  $  R $
 
is a normal subgroup of  $  X $)  
 
is a normal subgroup of  $  X $)  

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=49144
This article was adapted from an original article by O.N. Golovin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article