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 subgroup as a $V_{\mathfrak{U}}(G)$ and the corresponding marginal subgroup as $M_{\mathfrak{U}}(G)$. Note, that for the variety of all abelian groups $\mathfrak{A}$ (defined for the word $[x, y]$) we have $V_{\mathfrak{A}}(G) = G’$ and $M_{\mathfrak{A}}(G) = Z(G)$.
My question is:
Can the aforementioned statement be generalized to the following one:
Suppose $G$ is a finitely generated group and $\mathfrak{U}$ is a finitely based variety. Then $|V_{\mathfrak{U}}(G)| < \infty$ iff $[G:M_{\mathfrak{U}}(G)]<\infty$.
?
