Simultaneous triangularization over $\operatorname{GL}_n\left(\mathbb{F}_p\right)$

Lightning solution via Sylow

Deduce from
$$\bigl|\mathrm{GL}_n(\mathbb{F}_p)\bigr|=\prod_{k=0}^{n-1}\bigl(p^{n}-p^{k}\bigr)=p^{n(n-1)/2}\prod_{k=1}^{n}\bigl(p^{k}-1\bigr),$$the second factor being coprime to $p$, that $G$ is a Sylow $p$-subgroup. Since $U_n(\mathbb{F}_p)$ is another one, Sylow’s second theorem makes them conjugate, which settles the problem at once.