Let $p$ be a prime number . The order of a $p$-Sylow subgroup of the group $GL_{50}(F_p)$ of invertible $50\times50$ matrices with entries from the finite field $F_p$,equals which of the following
1.$p^{50}$
2.$p^{125}$
3.$p^{1250}$
4.$p^{1225}$
I think order of this group is $p^{2500}$ .A sylow p-subgroup of order $P^k$ divides order of the group but $p^{k+1}$ does not. I confused here how to apply these things .
Best Answer
Every element of $\operatorname{GL}_n(\mathbb{F}_p)$ is determined by the $n$ nonzero linearly independent vectors in $\mathbb{F}_p^n$ which form its columns. Counting the number of ways you can choose these reveals that the order of $\operatorname{GL}_n(\mathbb{F}_p)$ is $(p^n-1)(p^n-p)\cdots (p^n-p^{n-1})$. If $n=50$, then what is the power of $p$ dividing this order?