Kourovka Notebook Problem 4.40

Definitions and mathematical statement

Let $C$ be a fixed nontrivial group; the supplied example is $C={\mathbb{Z}}/2{\mathbb{Z}}$. Let $A$ and $B$ be arbitrary groups. Under the convention that a split extension of $B$ by $A$ has normal subgroup $B$ and quotient $A$, such an extension is a split exact sequence \[ 1\longrightarrow B\longrightarrow E\overset{p}{\longrightarrow}A\longrightarrow 1, \qquad s\colon A\longrightarrow E,\quad p\circ s=\operatorname{id}_A. \] After choosing the splitting and identifying the kernel with $B$, this gives $E\cong B\rtimes_\alpha A$, where $\alpha\colon A\to\operatorname{Aut}(B)$ is a homomorphism and \[ (b,a)(b',a')=(b\,\alpha(a)(b'),aa'). \] The source does not explicitly fix its convention for the phrase "extension of $B$ by $A$"; the sequence above makes the convention used in this draft explicit.

The notation $B\wr C$ denotes a wreath product, but the supplied record does not specify whether it is restricted or unrestricted. The regular restricted wreath product is $B^{(C)}\rtimes C$, where $B^{(C)}$ consists of functions $C\to B$ with finite support. The regular unrestricted wreath product is $B^C\rtimes C$, using all functions $C\to B$. In both cases multiplication in the base group is pointwise and the shift action is $(c\cdot f)(x)=f(c^{-1}x)$. These possibilities can differ when $C$ is infinite. Write $W=B\wr C$ for the wreath product intended by the cited source, and let $\operatorname{Aut}(W)$ be its group of automorphisms.

The supplied background attributes to Merzlyakov a construction embedding all split extensions, in a certain unified way, into $A\times\operatorname{Aut}(W)$. Under the extension convention above, denote the embeddings from that construction symbolically by \[ \iota_\alpha\colon B\rtimes_\alpha A\hookrightarrow A\times\operatorname{Aut}(W), \qquad \alpha\in\operatorname{Hom}(A,\operatorname{Aut}(B)). \] The problem asks for a description of how the image subgroups \[ \iota_\alpha(B\rtimes_\alpha A)\leq A\times\operatorname{Aut}(W) \] are situated in this ambient direct product, for arbitrary $A,B$ and fixed nontrivial $C$. The maps $\iota_\alpha$ are placeholders for the cited construction, not specified embeddings supplied by this draft. The record neither gives that construction nor defines the intended scope of "situated." Consequently, an exact characterization target cannot be recovered from the supplied record alone; the embedding construction, wreath-product convention, and intended subgroup relationships require clarification from the cited source.

Short English statement

For a fixed nontrivial group $C$, Merzlyakov's problem asks how the split extensions associated with arbitrary groups $A$ and $B$ occur inside $A\times\operatorname{Aut}(B\wr C)$ under a cited common embedding construction. The supplied record does not specify the construction, the wreath-product convention, or precisely which relationships among the embedded subgroups should be described.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $700248$, exact source problem code catalogue entry, title "Kourovka Notebook Problem 4.40." The registry labels the problem open; this is supplied metadata, not an independently verified mathematical assessment.
• Yu. I. Merzlyakov, proposer, 1973. The supplied record identifies the original source as The Kourovka Notebook, 4th issue (1973), Problem 4.40, and its later presentation in the 21st issue edition (2026), arXiv:1401.0300v48. The collection metadata names E. I. Khukhro and V. D. Mazurov as editors.
• Yu. I. Merzlyakov, Algebra and Logic, volume 9, no. 5 (1970), pp. 326--337. The supplied statement cites this reference for the unified embedding construction; no article title or construction details are supplied, and the reference has not been independently checked here.
• Supplied discussion: Achyuth Jayadevan, Preprint, 2026, https://zenodo.org/records/22727276. The record explicitly labels the asserted description an "unverified AI solution" claim, not an established solution.
• Supplied status review dated 2026-10-16: canonical status open, evidence category unrefereed_claim. Its note reports a $\#$ marker for the unverified AI solution claim in the October 2026 21st issue, arXiv:1401.0300v48. This is supplied dated curation, not independently verified source or literature assessment.