Kourovka Notebook Problem 21.10

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a finite set $X$, let $F(X)$ be the free group on $X$. A presentation $\langle X\mid R\rangle$ denotes $F(X)/\langle\!\langle R\rangle\!\rangle$, where $R$ is a finite set of relator words and double angle brackets denote normal closure. In the source's terminology a presentation is finite when its presented group is finite, and it is just finite if deleting any one specified relator produces an infinite group.

Statement recorded in the supplied source.

Does every finite group $G$ admit a presentation $\langle X\mid R\rangle\cong G$ such that \[ \left|F(X)/\langle\!\langle R\setminus\{r\}\rangle\!\rangle\right|=\infty \qquad\text{for every }r\in R? \] The words in the other relators and the generating set are kept fixed when a relator is removed. [S2] [S3]

Short English statement

Can every finite group be presented so that removal of any single relation makes the resulting group infinite?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2519 (KOU-21.10). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Y. Barnea, Problem 21.10, in E. I. Khukhro and V. D. Mazurov (eds.), The Kourovka Notebook: Unsolved Problems in Group Theory, 21st edition, arXiv:1401.0300v48 (6 October 2026). This version records an affirmative answer; the displayed question is the historically registered target. https://arxiv.org/abs/1401.0300v48.
• S3 Marc Lackenby, Every finite group admits a just finite presentation, arXiv:2605.10402v1 (2026), Introduction and Theorem 1. The theorem gives an affirmative answer with finite generating and relator sets. https://arxiv.org/abs/2605.10402v1.