Kourovka Notebook Problem 1.51

Definitions and mathematical statement

For a field $K$ and a positive integer $n$, let $\operatorname{GL}_n(K)$ denote the group of invertible $n\times n$ matrices over $K$. A matrix group over $K$ means a subgroup $G\leq\operatorname{GL}_n(K)$.

A strict total order $<$ on a group $G$ is left invariant if \[ \forall g,x,y\in G,\qquad x<y\Longrightarrow gx<gy. \] It is invariant on both sides if \[ \forall a,b,x,y\in G,\qquad x<y\Longrightarrow axb<ayb. \] The group is left-orderable if it admits a left-invariant strict total order, and bi-orderable if it admits a strict total order invariant on both sides.

Problem (A. I. Kokorin, 1965; Kourovka Notebook 1.51). The supplied question is: What conditions ensure a matrix group over a field (of complex numbers) to be orderable?

In the notation above, the requested task is to identify conditions on $G\leq\operatorname{GL}_n(K)$ ensuring the existence of the intended invariant total order. The supplied record does not specify whether "orderable" means left-orderable or bi-orderable. It also leaves unclear whether $K$ ranges over arbitrary fields, over subfields of ${\mathbb{C}}$, or is specifically ${\mathbb{C}}$. These alternatives are unresolved qualifications of the statement. The question does not specify whether the desired conditions are merely sufficient or constitute a necessary-and-sufficient characterization.

Short English statement

Kokorin asks for conditions ensuring that a matrix group admits an invariant total order. The supplied formulation leaves both the intended invariance of the order and the permitted fields unclear; it also does not specify whether a complete characterization is required.

Sources

• Local UnsolvedMath registry, permanent ID $700046$, source problem code catalogue entry, title Kourovka Notebook Problem 1.51. Supplied upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Supplied source attribution: The Kourovka Notebook (Unsolved Problems in Group Theory), original problem 1.51, first issue (1965); proposed by A. I. Kokorin.
• The supplied background identifies the twenty-first issue edition (2026), arXiv:1401.0300v48. The collection description credits E. I. Khukhro and V. D. Mazurov as editors. This attribution and edition information have not been independently verified here.
• The local record labels the problem open. This is a supplied registry status, not an independently verified assessment of the literature. No separate literature sources or literature-verification date are supplied.