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 $\prod_{j=1}^n (p^{n-j+1}-1)$.
But this doesn't seem to carry me any further... Please help :(