Kourovka Notebook Problem 19.56

Definitions and mathematical statement

For a finite group $G$, write $|g|$ for the order of an element $g\in G$, and use the convention $[x,y]=x^{-1}y^{-1}xy$. Define the set of $\star$-commutators by \[ S(G)=\left\{[x,y]:x,y\in G,\ |x|=p^r,\ |y|=q^s,\ p,q\text{ distinct primes},\ r,s\in{\mathbb{Z}}_{\geq 1}\right\}. \] Thus the two elements defining a $\star$-commutator have coprime prime-power orders; the resulting commutator itself need not have prime-power order and may be the identity. A group $G$ is soluble if its derived series, defined by \[ G^{(0)}=G,\qquad G^{(n+1)}=\langle [u,v]:u,v\in G^{(n)}\rangle, \] satisfies $G^{(m)}=\{1\}$ for some integer $m\geq 0$.

Question (V. S. Monakhov, Kourovka Notebook Problem 19.56). Is it true that every finite group $G$ satisfying \[ \forall a,b\in S(G),\qquad \gcd(|a|,|b|)=1\ \Longrightarrow\ |ab|\geq |a|\,|b| \] is soluble?

The supplied source does not specify a commutator convention; the convention above makes the notation explicit. Prime-power orders are interpreted as positive powers of primes. If order $1$ is also admitted under that terminology, the only additional possible $\star$-commutator value is the identity, and any inequality involving it is an equality. The supplied registry labels the question open and separately records an unverified affirmative AI solution claim; that claim is not treated here as an established solution.

Short English statement

Must a finite group be soluble if the order of the product of any two $\star$-commutators of coprime orders is at least the product of their orders? Here a $\star$-commutator comes from two elements of coprime prime-power orders. The supplied affirmative solution claim remains explicitly unverified.

Sources

• Local UnsolvedMath registry, permanent ID $701604$, exact source code catalogue entry, title Kourovka Notebook Problem 19.56; supplied complete source record, upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• V. S. Monakhov, Problem 19.56, The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018); also attributed in the supplied record to the 21st issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The supplied background and status review report an explicitly unverified affirmative AI solution claim attributed to M. Kinyon and J. Urban, Preprint (2026): https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/09/kourovka-mathematics.pdf. The supplied review says the claim is marked with $\#$ in the October 2026 version of the 21st issue.
• Supplied status review dated 2026-10-16: canonical status open, evidence classification unrefereed_claim. This is supplied dated curation, not an independently verified literature assessment; neither the cited notebook version nor the claimed solution has been independently checked here.