Kourovka Notebook Problem 21.72

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.

Use the source's definition of a Tarski monster: a finitely generated noncyclic group whose proper nontrivial subgroups are all isomorphic to each other. No restriction that these subgroups be cyclic is silently added. A discrete group is amenable if there is a positive linear functional $m:\ell^\infty(G)\to\mathbb R$ with $m(1)=1$ invariant under all left translations. A group is torsion-free when no nonidentity element has finite order, and has prime exponent $p$ when $g^p=1$ for every element and $G$ is nontrivial.

Statement recorded in the supplied source.

(a) Does there exist an amenable torsion-free Tarski monster in the specified sense?

(b) Does there exist an amenable Tarski monster of prime exponent? The prime is not fixed in advance. [S2]

Short English statement

Do amenable Tarski monsters exist, either torsion-free or of prime exponent?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2581 (KOU-21.72). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 D. Kielak, Problem 21.72, 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). https://arxiv.org/abs/1401.0300v48.