Kourovka Notebook Problem 21.134

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 group $G$, put $t_G(n)=|\{x\in G:x^n=1\}|$ for $n\geq1$. The solvable radical $R(G)$ is the largest solvable normal subgroup. A finite group $H$ is almost simple if $S\leq H\leq\operatorname{Aut}(S)$ for some nonabelian finite simple group $S$.

Statement recorded in the supplied source.

Suppose finite groups $G,H$ satisfy $t_G(n)=t_H(n)$ for every $n\geq1$. (a) If $R(H)=1$, must $R(G)=1$? (b) If $H$ is almost simple, must $G\cong H$? [S1][S2]

Short English statement

Ask whether the numbers of solutions to every power equation detect a trivial solvable radical and uniquely identify an almost simple group.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2643 (KOU-21.134). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 E. I. Khukhro and V. D. Mazurov (editors), The Kourovka Notebook, Unsolved Problems in Group Theory, No.21, version48 (6 October2026), Problem 21.134 and its remarks. https://arxiv.org/abs/1401.0300v48.