How to find the number $n_p$ of Sylow $p$-subgroups of $GL_n(\mathbb{F}_p)$? For example, in $GL_3(\mathbb{F}_p)$, $|GL_3(\mathbb{F}_p)|=(p^3-1)(p^3-p)(p^3-p^2) = p^3(p-1)^3(p+1)(p^2+p+1)$, we have the Heisenberg group $$ \begin{pmatrix} 1&x&y\\ 0&1&z\\ 0&0&1 \end{pmatrix} $$ as a Sylow $p$-subgroup. And its normalizer is at least the upper triangular matrices, which has order $p^3(p-1)^3$. So we now know $n_p\mid(p+1)(p^2+p+1)$. How to proceed from here to determine $n_p$?

  • $\begingroup$ You might look through Weir 1955 to see if he answers your question adequately... $\endgroup$ – Eric Towers Dec 4 '16 at 6:36
  • $\begingroup$ @EricTowers I see so $n_p = (p+1)(p^2+p+1)$ $\endgroup$ – snsunx Dec 4 '16 at 6:55

The book Groups and Representations by Alperin-Bell is a nice exposition for $\rm{GL}_n(\mathbb{F}_q)$. Here is a way to find the number of Sylow-$p$ subgroups in your question, whose details you can try to fill with the help of tools in the book mentioned.

(1) The group $\rm{U}_n(\mathbb{F}_p)$ of upper triangular matrices with all $1$ on diagonal is a Sylow-$p$ subgroup of $\rm{GL}_n(\mathbb{F}_p)$.

(2) The number of Sylow-$p$ subgroups in any group is equal to the index of normalizer of Sylow-$p$ subgroup.

(3) For the group $\rm{U}_n(\mathbb{F}_p)$, try to show that the group $B_n(\mathbb{F}_p)$ of upper triangular invertible matrices is the normalizer. For this, proceed as follows.

(3.0) Show that $\rm{U}_n(\mathbb{F}_p)$ is normal in $\rm{B}_n(\mathbb{F}_p)$. (For this, consider a map from $\rm{B}_n(\mathbb{F}_p)$ to itself, which sends any matrix to a diagonal matrix with same diagonal entries as in original; clear? Show that this is a homomorphism. Then what is kernel? Kernel is always a normal subgroup.)

(3.1) Suppose $g\in \rm{GL}_n(\mathbb{F}_p)$ normalizes $\rm{U}_n(\mathbb{F}_p)$.

(3.2) By Bruhat decompositon (follow book), $g$ can be written (uniquely) as $b_1wb_2$ where $b_1,b_2$ are in $\rm{B}_n(\mathbb{F}_p)$ and $w$ is a permutation matrix.

(3.3) Since $g=b_1wb_2$ and $b_1,b_2$ normalizes $\rm{U}_n(\mathbb{F}_p)$ (by (3.0)) hence $w$ normalizes $\rm{U}_n(\mathbb{F}_p)$.

(3.4) An easy marix computation shows that the only permutation matrix normalizing $\rm{U}_n(\mathbb{F}_p)$ is identity. Hence $g=b_1b_2\in \rm{B}_n(\mathbb{F}_p)$.

(3.5) Find the order of the normalizer $\rm{B}_n(\mathbb{F}_p)$ of Sylow-$p$ subgroup, and find its index in $\rm{GL}_n(\mathbb{F}_p)$. q.e.d


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