# The order of a $p$-Sylow subgroup of the group $GL_{50}(\mathbb{F}_p)$ of invertible $50×50$ matrices with entries from finite field $\mathbb{F}_p$

Let $p$ be a prime number. The order of a $p$-Sylow subgroup of the group $GL_{50}(\mathbb{F}_p)$ of invertible $50×50$ matrices with entries from finite field $\mathbb{F}_p$ equals:
$p^{50}$
$p^{125}$
$p^{1250}$
$p^{1225}$

I am completely stuck on it.can anyone help me please.

• To exclude some alternatives you may want to calculate the order of the subgroup of upper triangular matrices with 1s along the diagonal. – Jyrki Lahtonen Dec 16 '12 at 6:56

Because a matrix is invertible if and only if its columns are linearly independent, the order of $GL_{50}(\mathbb{F}_p)$ is
$$(p^{50}-1)(p^{50}-p)(p^{50}-p^2)\ldots(p^{50}-p^{49})$$
So you want to calculate the largest power of $p$ dividing this number. To do this, first find out what's the largest power of $p$ dividing $(p^{50} - p^k)$.
The answer is $p^{\frac {n(n-1)}{2}}$.just plug in $n=50$