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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 :(

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    Oh write, sorry didn't notice that. Editing.2012-11-12

1 Answers 1

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

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    Yes, that's it.2012-11-12