19.47 (2018)

Let $K$ be a finite extension of degree $n$ of a field $k$ of odd characteristic. The multiplicative group $K^*$ embeds into the group of all $k$-linear automorphisms $\text{Aut}_k(K)$ by the rule $t(x) = tx$ for all $x \in K$. For a fixed basis of $K$ over $k$, the group $\text{Aut}_k(K)$ is isomorphic to $\text{GL}(n, k)$. The image of $K^*$ is called a nonsplit maximal torus corresponding to the extension $K/k$ and is denoted by $T(K/k)$. A subgroup of $\text{GL}(n, k)$ is said to be rich in transvections if it contains all elementary transvections.

Let $H$ be a subgroup of $\text{GL}(n, k)$ which contains $T(K/k)$ and a one-dimensional transformation. Is $H$ rich in transvections?