Kourovka Notebook Problem 5.35

Definitions and mathematical statement

Let $F$ be a field and let $V$ be an $n$-dimensional vector space over $F$. Write $\operatorname{GL}(V)$ for the group of invertible $F$-linear maps $V\to V$, and write $I_V$ for the identity map. A transvection is an element $t\in\operatorname{GL}(V)$ such that \[ \operatorname{rank}(t-I_V)=1,\qquad (t-I_V)^2=0. \] Its residual line is $\operatorname{im}(t-I_V)$, and its fixed space is $\ker(t-I_V)$. A subgroup $G\leq\operatorname{GL}(V)$ is rich in transvections if $n\geq2$ and, for every hyperplane $H\leq V$ and every one-dimensional subspace $L\leq H$, there exists a transvection $t\in G$ with \[ \operatorname{im}(t-I_V)=L,\qquad \ker(t-I_V)=H. \] The source asks to describe the automorphisms of subgroups of $\operatorname{GL}_2(V)$ rich in transvections. Under the intended two-dimensional reading, the problem is: for every field $F$, every two-dimensional $F$-vector space $V$, and every subgroup $G\leq\operatorname{GL}(V)$ rich in transvections, describe all abstract group automorphisms of $G$, namely all bijections $\alpha:G\to G$ satisfying \[ \alpha(gh)=\alpha(g)\alpha(h)\qquad(g,h\in G). \] In dimension two, the richness condition is equivalently that every line $\ell\leq V$ occurs as both the residual line and the fixed space of some transvection in $G$. No restriction on the characteristic or cardinality of $F$ is supplied. The notation $\operatorname{GL}_2(V)$ is retained here as a source qualification: interpreting it as the linear group on a two-dimensional space, and interpreting automorphisms as abstract group automorphisms, require confirmation in mathematical review.

Short English statement

For each field and each two-dimensional vector space, describe all group automorphisms of every linear subgroup containing a transvection with each prescribed line as its fixed space and residual line. This uses the two-dimensional interpretation of the source notation; the supplied registry labels the problem open.

Sources

• Local UnsolvedMath registry, permanent ID $700318$, exact source problem code catalogue entry, title Kourovka Notebook Problem 5.35. The supplied record labels the problem open.
• The Kourovka Notebook (Unsolved Problems in Group Theory), fifth issue (1976), Problem 5.35; proposed by Yu. I. Merzlyakov.
• The supplied background also identifies the twenty-first issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors. These bibliographic details are supplied attribution and have not been independently verified here.