# Sylow p-subgroup of $GL_n(F_p)$

Here's the question:

Find a Sylow p-subgroup of $GL_n(F_p)$, and determine the number of Sylow p-subgroups.

So far here's what I've got:

• Order of $GL_n(F_p)$, which is $\prod_{j=1}^n p^n-p^{j-1}$ with j running from 1 to n,and thus the order of Sylow p-subgroup of it, and also its index.
• From the index, $\prod_{j=1}^n (p^{n-j+1}-1)$, we have the clue that the number of Sylow p-subgroups, s, both is congruent to 1 mod p and divides $\prod_{j=1}^n (p^{n-j+1}-1)$.

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Try to do this for $n=2$ first. Then for $n=3$. Then you'll see the pattern. (What is the order of the group, by the way?) – Mariano Suárez-Alvarez Nov 12 '12 at 5:36
$\prod(P^n-P^(j-1))$, with j running from 1 to n – Benjamin Lu Nov 12 '12 at 5:39
Edit the question and add that information there. (Also, notice that the numebr that you are claiming is the index is actually larger than the number you are claiming to be the order of the group!) – Mariano Suárez-Alvarez Nov 12 '12 at 5:41
Oh write, sorry didn't notice that. Editing. – Benjamin Lu Nov 12 '12 at 6:04

the order of sylow psubgroup is $p^\frac{n(n-1)}{2}$ a sylow p subgroup would be the subgroup of upper triangular matrices with diagonal entries 1