1
$\begingroup$

Fix a prime $p$.

How many Sylow $p$-groups are there in $GL_3(\Bbb F_p)$?

Let $s_p$ be the number of Sylow $p$-groups in $GL_3(\Bbb F_p)$. By the Sylow theorems, $s_p\equiv 1\mod p$ and if $p^e$ is the maximal prime power of the order of $GL_3(\Bbb F_p)$, then $s_p$ divides $\frac{\left|GL_3(\Bbb F_p) \right|}{p^e}$. Unfortunately I don't even really know how to figure out the cardinality of that group.

$\endgroup$
  • 1
    $\begingroup$ To compute the cardinality of the group, note that a general element of $GL_3(\mathbb F_p)$ is an invertible $3\times 3$ matrix over $\mathbb F_p$. The first column can be anything except $0$, so there are $p^3 - 1$ possibilities. The second column can be anything except a scalar multiple of the first column, so there are $p^3 - p$ possibilities. The third column can be anything except a linear combination of the first two columns, so there are $p^3 - p^2$ possibilities. So $|GL_3(\mathbb F_p)| = (p^3 - 1)(p^3 - p)(p^3 - p^2)$. $\endgroup$ – Bungo Jul 22 '16 at 18:09
  • $\begingroup$ The general formula for $|GL(n,q)|$ is computed the same way: see en.wikipedia.org/wiki/General_linear_group#Over_finite_fields $\endgroup$ – Bungo Jul 22 '16 at 18:09
4
$\begingroup$

I assume you use column vectors.

One particular Sylow $p$-subgroup is $$ L = \left\{ \begin{bmatrix} 1 & 0 & 0\\ a & 1 & 0\\ c & b & 1\\ \end{bmatrix} : a, b, c \in \Bbb F_p \right\}. $$ You may try and compute its normalizer. The index will give you the number of Sylow $p$-subgroups.

Alternatively, you may note that $x \in L$ iff $x - 1$ maps the underlying vector space to $\langle e_{2}, e_{3} \rangle$, $\langle e_{2}, e_{3} \rangle$ to $\langle e_{3} \rangle$ and $\langle e_{3} \rangle$ to $\{ 0 \}$, where $e_{1}, e_{2}, e_{3}$ is the standard basis. So there are as many Sylow $p$-subgroups as the number of pairs $U_{2} \supseteq U_{1}$ of subspaces, with $\dim(U_{i}) = i$. You will obtain the same number as with the previous method.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.