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].

14 questions
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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 ...
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 ...