Let $GL_n\mathbb{F}_p$ be the group of invertible $n \times n$ matrices with entries coming from $\mathbb{F_p}=\{0, 1, ..., p-1\}$ and with group operation multiplication of matrices. (We are writing $\mathbb{F_p}$ for this object to not confuse it with $\mathbb{Z_p}$, the $group$ of integers modulo $p$ with the operation $+$ modulo $p$; $\mathbb{F_p}$ is the $field$ of order $p$, where $p$ is prime.)
a) Show that $|GL_n\mathbb{F_p}|=(p^n-1)(p^n-p)(p^n-p^2)...(p^n-p^{n-1})$
Hint: You need to use the invertible nature of the matrices when its columns are linearly independent.
b) Show that $p \cdot p^2 \cdot ...\cdot p^{n-1}$ is the largest power of $p$ dividing the order of $GL_n\mathbb{F_p}$
c) Deduce that the matrices $A=(a_{ij})$ with $(a_{ij})=0$ when $i<j$ and $a_{ii}=1$ form a Sylow p-subgroup of $GL_n\mathbb{F_p}$.
I gather that we can think of $\mathbb{F_p}$ in the same way we think about $\mathbb{Q}, \mathbb{R}$ and $\mathbb{C}$. When we have vectors with entires in $\mathbb{F_p}$ we can talk about linear independence of such matrices in the usual way, and matrix multiplication as usual and them being invertible as usual, blah blah blah...
Just getting to grip with Sylow's theorems and this sort of algebra.