Kourovka Notebook Problem 21.115

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.

Let $n\geq1$ and $G$ be a finite group. A left coset of a subgroup $H$ is $gH=\{gh:h\in H\}$ and a right coset is $Hg=\{hg:h\in H\}$. Each $C_i$ below may be a left or right coset of its own subgroup, with no common subgroup, distinctness, or normality requirement. Put $U=\bigcup_{i=1}^n C_i$.

Statement recorded in the supplied source.

Whenever $U\ne G$, must \[ |G\setminus U|\geq\frac{|G|}{2^n}? \] The proposed lower bound is uniform over all finite groups and every selection of the $n$ cosets. [S2]

Short English statement

If n cosets fail to cover a finite group, is at least a fraction two-to-the-minus-n of the group left uncovered?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2624 (KOU-21.115). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 B. Sambale, Problem 21.115, 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.