# Questions tagged [verbal-subgroups]

A verbal subgroup of a group $G$, generated by the set of words $A \subset F_\infty$ ($F_\infty$ is a free group of countable rank) is a subgroup $V_A(G) = \langle \{h(w): w \in A, h \text{ is a homomorphism from } F_\infty \text{ to }G \} \rangle$. A verbal subgroup is always a characteristic one. To be used with the tag [group-theory].

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### Burnside groups with GAP system [closed]

My question is related to Burnside groups $B(n, 3)$ in the GAP system. I'm interested in ways to represent Burnside groups $B(n, 3)$ in GAP. The obvious representation using relations (see example for ...
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### Show that the group of all complex $2^n$th roots of unity, $n=0,1,2,...$, has a fully-invariant subgroup which is not verbal

This is the last part of Exercise 2.3.3 of Robinson's "A Course in the Theory of Groups (Second Edition)". The following are the other parts. Verbal subgroups of the group of all $2^n$th ...
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### Verbal Subgroups of $\mathbb{Z}^2$

So I want to try to show that all the verbal subgroups of an abelian group $G$ are of the form $G(X^n), n \geq 1$. I want to start with $\mathbb{Z}$ and $\mathbb{Z}^n$. So I worked with $\mathbb{Z}$, ...
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### Do there exist characteristic subgroups, that are not quasiverbal?

Let’s define a group quasiword as an element of $F_\infty \times P(F_\infty)$. Suppose $Q \subset F_\infty \times P(F_\infty)$ is a set of quasiwords. Define a quasiverbal subgroup of a group $G$ ...
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### Verbal subgroups of the group of all $2^n$th roots of unity.

Hi: Let $G$ be the multiplicative group of all complex $2^n$th roots of unity, $n=0,1,2,...$ Prove that $1$ and $G$ are the only verbal subgroups of $G$. Let $F$ be a free group on a countably finite ...
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### Is there some sort of formula for $[F_n : V_{\{x^4\}}(F_n)]$?

Suppose $F_n$ is a free group of rank $n$. It is a rather well known fact, that $b_4(n) = [F_n : V_{\{x^4\}}(F_n)]$ is finite for all $n \in \mathbb{N}$. Is there a some sort of formula for $b_4(n)$? ...
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### Is finite verbal subgroup equivalent to finite index of marginal subgroup?

There is a well known fact: If $G$ is a finitely generated group. Then $|G’| < \infty$ iff $[G:Z(G)]<\infty$. Suppose $\mathfrak{U}$ is a group variety. Let’s denote the corresponding verbal ...
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### Is there a formula for $[F_n : V_{\{x^3\}}(F_n)]$?

Suppose $F_n$ is a free group of rank $n$. It is a rather well known fact, that $b_3(n) = [F_n : V_{\{x^3\}}(F_n)]$ is finite for all $n \in \mathbb{N}$. Is there a some sort of formula for $b_3(n)$? ...
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### Are upper quasiverbal and lower quasiverbal subgroups always the same subgroup?

Let’s define a group quasiword as an element of $F_\infty \times P(F_\infty)$. Suppose $Q \subset F_\infty \times P(F_\infty)$ is a set of quasiwords. Define a prevariety described by $Q$ as a class ...
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### Do the quasiverbal subgroups always exist?

Let’s define a group quasiword as an element of $F_\infty \times P(F_\infty)$. Suppose $Q \subset F_\infty \times P(F_\infty)$ is a set of quasiwords. Define a quasivariety described by $Q$ as a class ...
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