Kourovka Notebook Problem 11.7

Definitions and mathematical statement

Let $G$ be a group. Define its lower central series by \[ \gamma_1(G)=G,\qquad \gamma_{n+1}(G)=[\gamma_n(G),G]\quad(n\geq 1), \] where $[H,G]$ is the subgroup generated by all commutators $h^{-1}g^{-1}hg$ with $h\in H$ and $g\in G$. Write \[ (*)\qquad \exists n\geq 1\;\;\gamma_n(G)\text{ is a free group}. \] Here a free group means a group having a free basis, with no restriction on the cardinality of that basis. Under the usual convention, the trivial group is free on the empty basis. The source does not specify a different convention or exclude $n=1$.

A group $G$ is residually finite if, for every $g\in G\setminus\{1\}$, there exist a finite group $F$ and a homomorphism $\varphi:G\to F$ such that $\varphi(g)\neq 1$.

1. (a) For a group given by a presentation $G=\langle X\mid R\rangle$, find conditions on that presentation under which \[ \left(\exists n\geq 1:\gamma_n(G)\text{ is free}\right) \Longrightarrow G\text{ is residually finite}. \] No finiteness assumption on $X$, $R$, or $G$ is supplied.
2. (b) Find conditions on the structure of a residually finite group $G$ that ensure \[ \exists n\geq 1:\gamma_n(G)\text{ is free}. \]

These are requests to identify hypotheses ensuring the indicated implications. The source does not specify a class of permitted hypotheses or require necessary and sufficient conditions.

Short English statement

Find presentation conditions ensuring that a group with a free term in its lower central series is residually finite. Conversely, find structural conditions ensuring that a residually finite group has such a free term. The scope of the requested conditions is unspecified.

Sources

• Local UnsolvedMath registry, supplied complete record: permanent ID $700700$, source problem code catalogue entry, title Kourovka Notebook Problem 11.7; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Supplied attribution: K. Bencsáth, proposed in 1990; The Kourovka Notebook (Unsolved Problems in Group Theory), original problem 11.7, 11th issue (1990). The supplied background also identifies the 21st issue edition (2026), arXiv:1401.0300v48. The collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The local record labels this problem open and records an update date of 2026-10-16. This is supplied registry information, not an independently verified assessment of mathematical or literature status.