Kourovka Notebook Problem 21.17

Definitions and mathematical statement

For a class $\mathcal X$ of groups, define four classes up to isomorphism. $H(\mathcal X)$ consists of all quotients of members of $\mathcal X$; $S(\mathcal X)$ consists of all their subgroups; $P(\mathcal X)$ consists of direct products indexed by arbitrary sets; and $P_f(\mathcal X)$ consists of products with finitely many factors. The arbitrary products are unrestricted Cartesian products, not direct sums. Operators are applied from the inside outward, so $S(H(\mathcal X))$ means subgroups of quotients, not quotients of subgroups.

A group $G$ is metabelian if its commutator subgroup $G'$ is abelian, or equivalently $G''=1$. For every class $\mathcal M$ consisting of metabelian groups, must the following inclusion hold? \[ H(S(P_f(\mathcal M)))\ \subseteq\ S(H(P(S(\mathcal M)))). \] More concretely, take finitely many groups $M_1,\ldots,M_t\in\mathcal M$, a subgroup $U\le\prod_{i=1}^tM_i$, and a normal subgroup $N\trianglelefteq U$. The question asks whether $U/N$ embeds into a quotient of a possibly infinite direct product of groups each isomorphic to a subgroup of a member of $\mathcal M$.

All choices of $U$ and $N$ are allowed, and the representing product and quotient may depend on them. The class is not assumed closed under any of these operations, and its groups need not be finite or finitely generated. The proposed inclusion is the target; no equality of the two operator expressions is being demanded.

Short English statement

For metabelian groups, can every quotient of a subgroup of a finite product embed in a quotient of an arbitrary product of subgroups?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2526 (KOU-21.17), Kourovka Notebook - New Problems, Issue 21. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 The Kourovka Notebook, 21st edition, new problems, Problem 21.17. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.