Verbal subgroup

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

$${\mathrm{\Gamma }}_1(G)=G\supseteq {\mathrm{\Gamma }}_2(G)\supseteq \cdots \supseteq {\mathrm{\Gamma }}_n(G)\supseteq \dots ,$$

where ${\mathrm{\Gamma }}_n(G)$ 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 $V(G)\varphi =V(G\varphi )$ is valid for any homomorphism $\varphi$. 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 $R⊲X$, $S⊲X$( here $R⊲X$ means that $R$ is a normal subgroup of $X$) and $R\subset S$, then $V(R)\subset V(S)$.

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