Kourovka Notebook Problem 21.106

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.

The group language has multiplication, inverse, and the identity constant. A first-order formula $\phi(x)$ has one free variable and no parameters from the group; other variables may be bound by quantifiers. Put $G_\phi=\{g\in G:G\models\phi(g)\}$ and $\phi(G)=\langle G_\phi\rangle$. A group $G$ is residually finite if for each $g\ne1$ some homomorphism from $G$ to a finite group has nonidentity image of $g$. A formula is concise on a class of groups when finiteness of $G_\phi$ always implies finiteness of $\phi(G)$.

Statement recorded in the supplied source.

Is every parameter-free first-order group formula with one free variable concise on the class of all residually finite groups? In symbols, must \[ \left(G\text{ residually finite and }|G_\phi|<\infty\right) \quad\Longrightarrow\quad|\langle G_\phi\rangle|<\infty \] hold for every such $\phi$ and every such $G$, with no finite-generation or torsion assumption? [S2]

Short English statement

Do finitely many values of any one-variable first-order formula always generate a finite subgroup inside a residually finite group?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2615 (KOU-21.106). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 M. Petschick, Problem 21.106, 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 a negative answer; the displayed question is the historically registered target. https://arxiv.org/abs/1401.0300v48.