This question has already been asked here. However, the answer there isn't particularly helpful.
Consider $G = GL_{n}(\mathbb{F}_{q})$ where $q = p^{r}$ where $p$ is prime. Show that the upper unipotent matrices (say $P$) are a p-Sylow subgroup of $G$.
Furthermore, find conditions so that every element of $P$ has order dividing $p$.
Attempt at a solution -
I recognize that this is true. However, is there an easy way to determine that it is? I know that the cardinality of $P$ is $q^{\frac{n(n-1)}{2}}$ and that the cardinality of $G$ is $(q^{n} - 1)(q^{n} - q) ... (q^{n} - q^{n-1})$ but there doesn't seem to be an easy way to factor the above equation.
For the second part, you can write any upper nilpotent matrix $A$ as $I + B$ where $B$ is an upper triangular matrix with $0's$ along the diagonal. It seems as though by the binomial theorem that $p \geq n$ and $r \geq p - n$ guarantees this. However, is this the exact characterization of when this property is true?
Edit - this is homework by the way.