6.38 (1978)

a) Let $k$ be a (commutative) field. Find all irreducible subgroups $G$ of $\text{GL}_n(k)$ having the property that $G \cap C \neq \varnothing$ for every conjugacy class $C$ of $\text{GL}_n(k)$. I conjecture that $G = \text{GL}_n(k)$ except in case $n = \text{char}\,k = 2$, the field $k$ is quadratically closed, and $G$ is conjugate to the group of all matrices of the form $\begin{pmatrix} \alpha & 0 \\ 0 & \beta \end{pmatrix}$, $\begin{pmatrix} 0 & \alpha \\ \beta & 0 \end{pmatrix}$ where $\alpha \neq 0$ and $\beta \neq 0$.
b) Is it true that an arbitrary subgroup of $GL_n(k)$ that intersects every conjugacy class is parabolic? How far is the same statement true for subgroups of other groups of Lie type?