21.35 (2026)

Let $G$ be a finite group, $w$ a multilinear commutator group-word, and $p$ a prime. Suppose that $p$ divides the order $|xy|$ whenever $x$ is a $w$-value of $p'$-order in $G$ and $y$ is a $w$-value in $G$ of order divisible by $p$. Is it true that then the verbal subgroup $w(G)$ must be $p$-nilpotent?

Without the assumption that $w$ be multilinear, the answer is negative. An affirmative answer has been obtained in several special cases (J. Algebra, 609 (2022), 926–936).