View source for Verbal subgroup

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The subgroup  $  V( G) $
of a group  $  G $
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 } \} $, 
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|verbal congruence]] (see also [[Algebraic systems, variety of|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|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====
<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>