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

Let $n$ be a positive integer and let $p$ be a prime. Let $G$ be a subgroup of $\operatorname{GL}_n\left(\mathbb{F}_p\right)$ of order $p^{n(n-1) / 2}$ . Show that the elements of $G$ are simultaneously triangularizable.

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References

Rached Mneimné — Réduction des endomorphismes

Details

Proposition 13.7.A, page 73

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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.

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