21.17 (2026)

If $\mathfrak{X}$ is a class of groups, let $\mathbf{H}(\mathfrak{X})$ denote the class of homomorphic images of groups in $\mathfrak{X}$, let $\mathbf{S}(\mathfrak{X})$ denote the class of groups isomorphic to subgroups of groups in $\mathfrak{X}$, let $\mathbf{P}(\mathfrak{X})$ denote the class of groups isomorphic to (unrestricted) direct products of families of groups in $\mathfrak{X}$, and let $\mathbf{P}_f(\mathfrak{X})$ denote the class of groups isomorphic to direct products of finite families of groups in $\mathfrak{X}$. By Birkhoff’s theorem, $\mathbf{H}(\mathbf{S}(\mathbf{P}(\mathfrak{X})))$ is the variety of groups generated by $\mathfrak{X}$.

If $\mathfrak{M}$ is a class of metabelian groups, must $\mathbf{H}(\mathbf{S}(\mathbf{P}_f(\mathfrak{M}))) \subseteq \mathbf{S}(\mathbf{H}(\mathbf{P}(\mathbf{S}(\mathfrak{M}))))$? This is Question 27 in (G. M. Bergman, Algebra Universalis, 26 (1989), 267–283).