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:

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

  • $\begingroup$ To exclude some alternatives you may want to calculate the order of the subgroup of upper triangular matrices with 1s along the diagonal. $\endgroup$ – 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


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 $n=50$


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