Kourovka Notebook Problem 21.38

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 $G$, let $s(G)$ be the largest nonnegative integer $k$ for which \[ \forall x_1,\ldots,x_k\in G\setminus\{1\}\ \exists y\in G:\quad \langle x_i,y\rangle=G\quad(1\leq i\leq k), \] with $s(G)=\infty$ if this holds for every $k$. Here $\langle\cdot\rangle$ denotes the subgroup generated by the listed elements; repeated $x_i$ are permitted.

Statement recorded in the supplied source.

Does there exist a group $G$ with spread $s(G)=1$? Explicitly, does some group satisfy \[ \forall x\ne1\ \exists y:\langle x,y\rangle=G, \] while having nonidentity elements $u,v$ such that \[ \forall y\in G:\quad \langle u,y\rangle\ne G\ \text{or}\ \langle v,y\rangle\ne G? \] [S2]

Short English statement

Does there exist a group where every nonidentity element has a generating partner, but some pair of nonidentity elements has no common generating partner?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2547 (KOU-21.38). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 S. Harper, C. Donoven and S. Corson, Problem 21.38, 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.