Kourovka Notebook Problem 17.80

Definitions and mathematical statement

For elements $a,b$ of a group, adopt the commutator convention $[a,b]=a^{-1}b^{-1}ab$, and define the iterated commutator recursively by \[ [a,{}_1b]=[a,b],\qquad [a,{}_{k+1}b]=[[a,{}_kb],b]\quad(k\geq 1). \] Thus $[a,{}_nb]$ is the left-normed commutator with $n$ occurrences of $b$. The supplied record uses iterated commutators but does not explicitly specify the commutator convention or bracketing; these conventions are an explicit interpretation requiring confirmation against the source.

For each integer $n\geq 3$, let \[ M_n=\left\langle x,y\ \middle|\ x=[x,{}_ny],\quad y=[y,{}_nx]\right\rangle. \] This is a group presentation, with no additional relations assumed. Is it true that \[ \forall n\in{\mathbb{Z}},\qquad n\geq 3\ \Longrightarrow\ |M_n|=\infty? \] The supplied statement also refers to Kourovka Notebook Problem 11.18, whose contents are not supplied.

Short English statement

For every integer $n>2$, consider the group generated by $x$ and $y$, subject to each generator equalling its iterated commutator with $n$ copies of the other generator. Must every such group be infinite? The explicit commutator conventions used here require source confirmation.

Sources

• Local UnsolvedMath registry, permanent ID $701382$, exact source code catalogue entry, title Kourovka Notebook Problem 17.80. Supplied upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82; record created and updated on 2026-10-16T14:00:00Z.
• Attributed in the supplied record to O. Macedońska, proposed in 2010. Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), Problem 17.80; the record also identifies the 21st issue edition (2026), arXiv:1401.0300v48. The collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The supplied statement includes the cross-reference "See also 11.18." The referenced problem text is not included.
• The local registry labels this problem open. This is a supplied status label, not an independently verified assessment of the literature. The cited source has not been independently consulted for this draft.