Kourovka Notebook Problem 21.148

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.

A right order on $G$ is a total order $<$ satisfying $x<y\Rightarrow xg<yg$ for all $g\in G$. A subgroup $H$ is convex for this order if $h_1<x<h_2$, with $h_1,h_2\in H$, implies $x\in H$. Let $\mathrm{RC}(G)$ consist of subgroups convex for at least one right order; different subgroups may use different orders. Order $\mathrm{RC}(G)$ by inclusion. Write $\wedge,\vee$ for the intrinsic meet and join in this lattice. Distributivity means both $H\wedge(K\vee L)=(H\wedge K)\vee(H\wedge L)$ and its dual; a chain means any two elements are comparable.

Statement recorded in the supplied source.

For every right-orderable $G$, is the lattice $\mathrm{RC}(G)$ distributive if and only if it is a chain? [S1][S2]

Short English statement

Is distributivity of the lattice of right-relatively convex subgroups equivalent to their being totally ordered by inclusion?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2657 (KOU-21.148). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 E. I. Khukhro and V. D. Mazurov (editors), The Kourovka Notebook, Unsolved Problems in Group Theory, No.21, version48 (6 October2026), Problem 21.148 and its remarks. https://arxiv.org/abs/1401.0300v48.