Let $p$ be a prime and $G\subset\operatorname{GL}_n(\mathbb{F}_p)$ be a subgroup. I wondering about the following question:
Is the map $\operatorname{Tr}:G\to\mathbb{F}_p$ surjective?
I know it's true for $G=\operatorname{SL}_n(\mathbb{F}_p)$ with $n>1$. Is it also true for $O_n(q,\mathbb{F}_p)$ where $q$ is a non-degenerate quadratic form over $\mathbb{F}_p$? Is it true for every finite group of Lie type if $p$ is large enough?