Kourovka Notebook Problem 21.87

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 $G$, write $d(G)$ for its minimal number of generators, and $[G:H]=|G|/|H|$ for a subgroup index. A family of indices has no common divisor when its greatest common divisor is $1$. All subgroups are ordinary abstract subgroups and no normality is required.

Statement recorded in the supplied source.

For every finite group $G$ and integer $d\geq0$, if there are subgroups $H_1,\ldots,H_m\leq G$ with $d(H_i)\leq d$ for every $i$ and \[ \gcd([G:H_1],\ldots,[G:H_m])=1, \] must $d(G)\leq d+1$? [S2]

Short English statement

If a finite group has d-generator subgroups with relatively prime indices collectively, must the whole group have at most d+1 generators?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2596 (KOU-21.87). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 A. Lucchini, Problem 21.87, 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.