Kourovka Notebook Problem 21.32

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 group $H$, its derived subgroup $H'=[H,H]$ is generated by all commutators $x^{-1}y^{-1}xy$. A finite group $G$ is integrable if there exists a finite group $H$ with $H'\cong G$. A finite multiplication table is one explicit encoding of the input group; any proposed complexity bound must specify its chosen encoding and size parameter.

Statement recorded in the supplied source.

Is there an algorithm that, on every input finite group $G$, terminates and correctly decides whether some finite group $H$ satisfies $H'\cong G$? If such a decision algorithm exists, determine the complexity of this decision problem, with the input representation and size measure made explicit. [S2] [S3]

Short English statement

Decide whether a finite group occurs as the commutator subgroup of a finite group, and determine the computational complexity of doing so.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2541 (KOU-21.32). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 P. J. Cameron, Problem 21.32, 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 decidability and a complexity result; the displayed question is the historically registered target. https://arxiv.org/abs/1401.0300v48.
• S3 Sathasivam Kalithasan and Viji Z. Thomas, On the decidability of the integrability of finite groups, arXiv:2602.18829v1 (2026), Introduction, main theorem and corollary. It proves decidability by a computable finite order bound. https://arxiv.org/abs/2602.18829v1.