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How many matrices are in $GL_n(\mathbb{F}_p)$ are upper triangular? How many upper triangular matrices with 1 on the diagonal are there?

$\textbf{proof.}$ Let $U$ be the subgroup of upper triangular matrices of $GL_n(\mathbb{F}_p)$ and let $D$ be the subgroup of upper triangular matrices with diagonal entries 1.

To find the order of $U$, we note that the diagonal entries can never by zero, since if one diagonal entry was zero the resulting matrix wouldn't be invertible. Also a row in an invertible upper triangular matrix will never be a multiple of a row above it, since the diagonal entry of the preceding row can only be zeroed out by multiplying the row by zero. Thus $$ \lvert U \rvert = (p-1)p^{n-1} \cdot (p-1)p^{n-2} \cdot \ldots \cdot (p-1) = (p-1)^n p^{1 + 2 + \ldots + n-1} = (p-1)^n p^{n(n-1)/2}. $$

To find the order of $D$ we divide out the nonzero diagonal entries $$ \lvert D \rvert = \frac{\lvert U \rvert}{(p-1)^n} = p^{n(n-1)/2}. $$ //

Is this correct and is it articulated alright? If there are other ways of counting this I'd like to know as well, thanks.

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    $\begingroup$ $U$ is a Sylow $p$-subgroup. See also this question. $\endgroup$ Jan 21, 2018 at 20:50
  • $\begingroup$ The proof is correct. You don't need the "Also a row in an invertible upper triangular matrix will never be a multiple of a row above it" sentence, though. $\endgroup$ Nov 14, 2018 at 0:03

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