Kourovka Problem 21.8 — horizontal class transpositions

Kourovka Problem $21.8$, proposed by V. G. Bardakov and A. L. Iskra, asks whether the group generated by horizontal class transpositions of moduli through $k$ is $S_{\operatorname{lcm}(2,\ldots,k)}$ for every $k\geq4$. Two independent 2026 arXiv notes report affirmative proofs.

Known results

• Earlier work (2024) established possible orders of products of two horizontal class transpositions as $\{1,2,3,4,6,12\}$, verified computationally, and recorded the full conjecture.

2026 proofs

A July 2026 paper proves that, with $L_k=\operatorname{lcm}(2,3,\ldots,k)$, $\operatorname{CT}_{(k)}\cong S_{L_k}$ for every $k\geq4$; it also reports that Pan obtained an independent proof using multiple transitivity. A separate April 2026 note proves the equivalent statement for every $n>3$.

Current status (as of July 2026): The conjecture is resolved by arXiv proofs: $\operatorname{CT}_{(k)}\cong S_{\operatorname{lcm}(2,\ldots,k)}$ for every $k\geq4$.