Kourovka Notebook Problem 21.75

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 $0\leq r<m$, let $r(m)=r+m\mathbb Z$. If $r_1(m_1)$ and $r_2(m_2)$ are disjoint, their class transposition sends $r_1+km_1$ to $r_2+km_2$ and vice versa for each $k\in\mathbb Z$, fixing all other integers. Let $\operatorname{CT}(\mathbb Z)$ be generated by all these permutations. For a set $P$ of odd primes, $\operatorname{CT}_P(\mathbb Z)$ is generated by the class transpositions for which each prime factor of both moduli belongs to $P\cup\{2\}$. The notation $\langle A,B\rangle$ denotes the subgroup generated jointly by two subgroups. The prime sets may be infinite.

Statement recorded in the supplied source.

For every two sets $P_1,P_2$ of odd primes with $P_1\nsubseteq P_2$ and $P_2\nsubseteq P_1$, is the natural inclusion always strict: \[ \langle\operatorname{CT}_{P_1}(\mathbb Z),\operatorname{CT}_{P_2}(\mathbb Z)\rangle \lneq\operatorname{CT}_{P_1\cup P_2}(\mathbb Z)? \] [S2]

Short English statement

Do two incomparable prime sets always generate a proper subgroup of the class-transposition group permitted by their union?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2584 (KOU-21.75). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 S. Kohl, Problem 21.75, 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). See also source definitions in Problems 17.57 and17.60. This version records equality instead of the proposed strict containment; the displayed question is the historically registered target. https://arxiv.org/abs/1401.0300v48.