Kourovka Notebook Problem 21.86

Definitions and mathematical statement

For a positive integer $n$, let $S_n$ act on $[n]=\{1,\ldots,n\}$, and use normalized Hamming distance \[ d_n(\sigma,\tau)=\frac{1}{n}|\{i\in[n]:\sigma(i)\ne\tau(i)\}|. \] A permutation is fixed-point-free if $\sigma(i)\ne i$ for every $i\in[n]$. Let $G$ be an arbitrary group, with identity $1$. In the source's finite-approximation convention, $G$ is sofic if the following holds: for every finite subset $F\subseteq G$ containing $1$ and every real $\varepsilon>0$, there exist $n\ge1$ and a map $\phi:F\to S_n$ such that \[ \phi(1)=\mathrm{id}_{[n]},\qquad d_n(\phi(gh),\phi(g)\phi(h))<\varepsilon \quad(g,h,gh\in F), \] and $\phi(g)$ is fixed-point-free for every $g\in F\setminus\{1\}$.

Is every group sofic? This asks for the displayed approximations separately for every finite test set and every positive accuracy. The number of points $n$ and the map $\phi$ may depend on both $F$ and $\varepsilon$. The test set need not be a subgroup or be closed under multiplication: only products remaining inside it are tested. No compatibility between approximations for different test sets is required. The statement includes groups that are not finitely generated and does not assume a presentation or a countability hypothesis.

Short English statement

Can the multiplication on every finite part of any group be approximated arbitrarily accurately by finite permutations separating nonidentity elements from the identity?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2595 (KOU-21.86), Kourovka Notebook - New Problems, Issue 21. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 The Kourovka Notebook, 21st edition, version 47 (30 September 2026), new problems, Problem 21.86. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.