Kourovka Notebook Problem 1.87

Definitions and mathematical statement

For a field $K$ and a positive integer $n$, let \[ \operatorname{GL}_n(K)=\{A\in M_n(K):\det(A)\ne 0\}, \] where $M_n(K)$ denotes the set of $n\times n$ matrices over $K$. In the usual finite-dimensional sense, a matrix group over $K$ is a subgroup of $\operatorname{GL}_n(K)$ for some $n\ge 1$.

The supplied statement of Problem 1.87 is: "The same question for matrix groups (at least over a field of characteristic 0)." Thus the record asks to extend an earlier question to matrix groups, with fields satisfying $\operatorname{char}(K)=0$ explicitly identified as a case of interest.

The earlier question is absent from the supplied record. Consequently, the property being investigated, its hypotheses, and its quantifiers cannot be recovered, and no complete mathematical assertion can be stated from this record alone. The definitions above explain standard terminology; they do not supply the missing question or establish its intended scope. The parenthetical qualification does not resolve whether other characteristics are also intended.

Short English statement

A. L. Shmel'kin asks an unspecified earlier question for matrix groups, at least over fields of characteristic $0$. The missing antecedent prevents a complete mathematical formulation. The supplied registry labels the problem open; this label is not an independently verified mathematical or literature assessment.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $700076$, exact source code catalogue entry, title Kourovka Notebook Problem 1.87; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The supplied record labels its status as open.
• The record attributes Problem 1.87 to A. L. Shmel'kin, proposed in 1965, and cites The Kourovka Notebook (Unsolved Problems in Group Theory), first issue (1965), as reproduced in the 21st-issue edition (2026), arXiv:1401.0300v48. Its collection description names E. I. Khukhro and V. D. Mazurov as editors. These bibliographic details are supplied attribution and have not been independently verified.
• The supplied source excerpt contains only the dependent sentence reproduced above; it does not include the earlier question to which "the same question" refers.