Kourovka Notebook Problem 5.38

Definitions and mathematical statement

A group $G$ is finitely generated if $G=\langle S\rangle$ for some finite subset $S\subseteq G$. Define its derived series by $G^{(0)}=G$ and \[ G^{(n+1)}=[G^{(n)},G^{(n)}]\qquad(n\geq 0), \] where $[H,H]$ is the subgroup generated by the commutators $x^{-1}y^{-1}xy$ with $x,y\in H$. The group $G$ is soluble if $G^{(d)}=\{1_G\}$ for some integer $d\geq 0$. It is Hopfian if every surjective group homomorphism $G\to G$ is injective, and hence an automorphism.

The direct product $A\times B$ has multiplication $(a,b)(a',b')=(aa',bb')$.

Question (P. M. Neumann, 1976; Kourovka Notebook Problem 5.38). For every pair of finitely generated soluble Hopfian groups $A$ and $B$, is $A\times B$ Hopfian? Equivalently, must every surjective group homomorphism \[ \varphi:A\times B\longrightarrow A\times B \] be injective?

The supplied registry labels this problem open. Its accompanying discussion reports an affirmative answer only as an unverified AI solution claim attributed to Achyuth Jayadevan. That claim is not treated here as an established resolution.

Short English statement

If two groups are each finitely generated, soluble, and Hopfian, must their direct product also be Hopfian? The supplied source reports an unverified affirmative solution claim while retaining an open registry status.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $700320$, exact source problem code catalogue entry, titled Kourovka Notebook Problem 5.38.
• P. M. Neumann, Problem 5.38, The Kourovka Notebook (Unsolved Problems in Group Theory), 5th issue (1976), as identified in the supplied record. The record cites the 21st issue edition (2026), arXiv:1401.0300v48, and identifies E. I. Khukhro and V. D. Mazurov as editors of the collection.
• Achyuth Jayadevan, Preprint (2026), https://zenodo.org/records/22729301. The supplied discussion labels this an unverified AI solution claim giving an affirmative answer; it supplies no preprint title or verified proof assessment.
• Supplied status review dated 2026-10-16: canonical status open, evidence category unrefereed_claim. Its notes report a $\#$ marker for the unverified AI solution claim in the October 2026 21st issue, arXiv:1401.0300v48. This is supplied dated curation, not an independently verified literature assessment.