Meditation on the Sylow theorems II

In Part I we discussed some conceptual proofs of the Sylow theorems. Two of those proofs involve reducing the existence of Sylow subgroups to the existence of Sylow subgroups of Image may be NSFW.
Clik here to view. S_n and Image may be NSFW.
Clik here to view. $GL_n(\mathbb{F}_p)$ respectively. The goal of this post is to understand the Sylow Image may be NSFW.
Clik here to view.-subgroups of Image may be NSFW.
Clik here to view. $GL_n(\mathbb{F}_p)$ in more detail and see what we can learn from them about Sylow subgroups in general.

Explicit Sylow theory for Image may be NSFW.
Clik here to view. $GL_n(\mathbb{F}_p)$

Our starting point is the following.

Baby Lie-Kolchin: Let Image may be NSFW.
Clik here to view. be a finite Image may be NSFW.
Clik here to view.-group acting linearly on a finite-dimensional vector space Image may be NSFW.
Clik here to view. over Image may be NSFW.
Clik here to view. $\mathbb{F}_p$ . Then Image may be NSFW.
Clik here to view. fixes a nonzero vector; equivalently, Image may be NSFW.
Clik here to view. has a trivial subrepresentation.

Proof. If Image may be NSFW.
Clik here to view. $\dim V = n$ then there are Image may be NSFW.
Clik here to view. p^n - 1 nonzero vectors in Image may be NSFW.
Clik here to view. $V \setminus \{ 0 \}$ , so by the PGFPT Image may be NSFW.
Clik here to view. fixes at least one of them (in fact at least Image may be NSFW.
Clik here to view. p-1 but these are just given by scalar multiplication). Image may be NSFW.
Clik here to view. $\Box$

Now we can argue as follows. If Image may be NSFW.
Clik here to view. is a finite Image may be NSFW.
Clik here to view.-group acting on an Image may be NSFW.
Clik here to view.-dimensional vector space Image may be NSFW.
Clik here to view. over Image may be NSFW.
Clik here to view. $\mathbb{F}_p$ (equivalently, up to isomorphism, a finite Image may be NSFW.
Clik here to view.-subgroup of Image may be NSFW.
Clik here to view. $GL_n(\mathbb{F}_p)$ ), it fixes some nonzero vector Image may be NSFW.
Clik here to view. $v_1 \in V$ . Writing Image may be NSFW.
Clik here to view. $V_1 = \text{span}(v_1)$ , we get a quotient representation on Image may be NSFW.
Clik here to view. V/V_1 , on which Image may be NSFW.
Clik here to view. fixes some nonzero vector, which we lift to a vector Image may be NSFW.
Clik here to view. $v_2 \in V$ , necessarily linearly independent from Image may be NSFW.
Clik here to view. v_1 , such that Image may be NSFW.
Clik here to view. acts upper triangularly on Image may be NSFW.
Clik here to view. $V_2 = \text{span}(v_1, v_2)$ .

Continuing in this way we get a sequence Image may be NSFW.
Clik here to view. $v_1, \dots v_n$ of linearly independent vectors (hence a basis of Image may be NSFW.
Clik here to view.) and an increasing sequence Image may be NSFW.
Clik here to view. $V_k = \text{span}(v_1, \dots v_k)$ of subspaces of Image may be NSFW.
Clik here to view. that Image may be NSFW.
Clik here to view. leaves invariant, satisfying the additional condition that Image may be NSFW.
Clik here to view. fixes Image may be NSFW.
Clik here to view. $v_k \bmod V_{k-1}$ . The subspaces Image may be NSFW.
Clik here to view. V_k form a complete flag in Image may be NSFW.
Clik here to view., and writing the elements of Image may be NSFW.
Clik here to view. as matrices with respect to the basis Image may be NSFW.
Clik here to view. $v_1, \dots v_n$ , we see that the conditions that Image may be NSFW.
Clik here to view. leaves Image may be NSFW.
Clik here to view. V_k invariant and fixes Image may be NSFW.
Clik here to view. $v_k \bmod V_{k-1}$ says exactly that Image may be NSFW.
Clik here to view. acts by upper triangular matrices with Image may be NSFW.
Clik here to view.s on the diagonal in this basis.

Conjugating back to the standard basis, we’ve proven:

Proposition: Every Image may be NSFW.
Clik here to view.-subgroup of Image may be NSFW.
Clik here to view. $GL_n(\mathbb{F}_p)$ is conjugate to a subgroup of the unipotent subgroup Image may be NSFW.
Clik here to view. $U_n(\mathbb{F}_p)$ .

This is almost a proof of Sylow I and II for Image may be NSFW.
Clik here to view. $GL_n(\mathbb{F}_p)$ (albeit at the prime Image may be NSFW.
Clik here to view. only), but because we defined Sylow Image may be NSFW.
Clik here to view.-subgroups to be Image may be NSFW.
Clik here to view.-subgroups having index coprime to Image may be NSFW.
Clik here to view., we’ve only established that Image may be NSFW.
Clik here to view. $U_n(\mathbb{F}_p)$ is maximal, not that it’s Sylow.

We can show that it’s Sylow by explicitly dividing its order into the order of Image may be NSFW.
Clik here to view. $GL_n(\mathbb{F}_p)$ but there’s a more conceptual approach that will teach us more. Previously we proved the normalizer criterion: a Image may be NSFW.
Clik here to view.-subgroup Image may be NSFW.
Clik here to view. of a group Image may be NSFW.
Clik here to view. is Sylow iff the quotient Image may be NSFW.
Clik here to view. N_G(P)/P has no elements of order Image may be NSFW.
Clik here to view..

Claim: The normalizer of Image may be NSFW.
Clik here to view. $U = U_n(\mathbb{F}_p)$ is the Borel subgroup Image may be NSFW.
Clik here to view. $B = B_n(\mathbb{F}_p)$ of upper triangular matrices (with no restrictions on the diagonal). The quotient Image may be NSFW.
Clik here to view. B/U is the torus Image may be NSFW.
Clik here to view. $(\mathbb{F}_p^{\times})^n$ and in particular has no elements of order Image may be NSFW.
Clik here to view..

Corollary (Explicit Sylow I and II for Image may be NSFW.
Clik here to view. $GL_n(\mathbb{F}_p)$ : Image may be NSFW.
Clik here to view. $U_n(\mathbb{F}_p)$ is a Sylow Image may be NSFW.
Clik here to view.-subgroup of Image may be NSFW.
Clik here to view. $GL_n(\mathbb{F}_p)$ .

Proof. The normalizer Image may be NSFW.
Clik here to view. N_G(U) is the stabilizer of Image may be NSFW.
Clik here to view. $G = GL_n(\mathbb{F}_p)$ acting on the set of conjugates of Image may be NSFW.
Clik here to view.. We want to show that Image may be NSFW.
Clik here to view. N_G(U) = B , which would mean that the action of Image may be NSFW.
Clik here to view. on the conjugates of Image may be NSFW.
Clik here to view. can be identified with the quotient Image may be NSFW.
Clik here to view. G/B .

This quotient is the complete flag variety: it can be identified with the action of Image may be NSFW.
Clik here to view. on the set of complete flags in Image may be NSFW.
Clik here to view. $\mathbb{F}_p^n$ , since the action on flags is transitive and the stabilizer of the standard flag

Image may be NSFW.
Clik here to view. $\displaystyle 0 \subset \text{span}(e_1) \subset \text{span}(e_1, e_2) \subset \dots \subset \mathbb{F}_p^n$

is exactly Image may be NSFW.
Clik here to view.. So it suffices to exhibit a Image may be NSFW.
Clik here to view.-equivariant bijection between conjugates of Image may be NSFW.
Clik here to view. and complete flags which sends Image may be NSFW.
Clik here to view. to the standard flag, since then their stabilizers must agree.

But this is clear: given a complete flag

Image may be NSFW.
Clik here to view. $\displaystyle 0 = V_0 \subset V_1 \subset \dots \subset V_n = \mathbb{F}_p^n$

we can consider the subgroup of Image may be NSFW.
Clik here to view. $g \in G$ which preserves the flag (so Image may be NSFW.
Clik here to view. g V_i = V_i ) and which has the additional property that the induced action on Image may be NSFW.
Clik here to view. $V_{i+1}/V_i$ is trivial for every Image may be NSFW.
Clik here to view.. This produces Image may be NSFW.
Clik here to view. when applied to the standard flag, so produces conjugates of Image may be NSFW.
Clik here to view. when applied to all flags. In the other direction, given a conjugate Image may be NSFW.
Clik here to view. $gUg^{-1}$ of Image may be NSFW.
Clik here to view., it has a Image may be NSFW.
Clik here to view.-dimensional invariant subspace Image may be NSFW.
Clik here to view. V_1 acting on Image may be NSFW.
Clik here to view. $\mathbb{F}_p^n$ , quotienting by this subspace produces a unique Image may be NSFW.
Clik here to view.-dimensional invariant subspace Image may be NSFW.
Clik here to view. V_2/V_1 acting on Image may be NSFW.
Clik here to view. $\mathbb{F}_p^n/V_1$ , etc.; this produces the standard flag when applied to Image may be NSFW.
Clik here to view., so produces Image may be NSFW.
Clik here to view. applied to the standard flag when applied to conjugates of Image may be NSFW.
Clik here to view.. So we get our desired Image may be NSFW.
Clik here to view.-equivariant bijection between conjugates of Image may be NSFW.
Clik here to view. and complete flags, establishing Image may be NSFW.
Clik here to view. N_G(U) = B as desired. Image may be NSFW.
Clik here to view. $\Box$

(This argument works over any field.)

From here it’s not hard to also prove

Explicit Sylow III for Image may be NSFW.
Clik here to view. $GL_n(\mathbb{F}_p)$ : The number Image may be NSFW.
Clik here to view. n_p of conjugates of Image may be NSFW.
Clik here to view. $U_n(\mathbb{F}_p)$ in Image may be NSFW.
Clik here to view. $GL_n(\mathbb{F}_p)$ divides the order of Image may be NSFW.
Clik here to view. $GL_n(\mathbb{F}_p)$ and is congruent to Image may be NSFW.
Clik here to view. $1 \bmod p$ .

Proof. Actually we can compute Image may be NSFW.
Clik here to view. n_p exactly: we established above that it’s the number of complete flags in Image may be NSFW.
Clik here to view. $\mathbb{F}_p^n$ (on which Image may be NSFW.
Clik here to view. $GL_n(\mathbb{F}_p)$ acts transitively, hence the divisibility relation), and a classic counting argument (count the number of possibilities for Image may be NSFW.
Clik here to view. V_1 , then for Image may be NSFW.
Clik here to view. V_2 , etc.) gives the Image may be NSFW.
Clik here to view.-factorial

Image may be NSFW.
Clik here to view. $\displaystyle n_p = [n]_p! = \prod_{i=1}^n [i]_p = \prod_{i=1}^n \left( p^{i-1} + p^{i-2} + \dots + 1 \right)$

which is clearly congruent to Image may be NSFW.
Clik here to view. $1 \bmod p$ . Image may be NSFW.
Clik here to view. $\Box$

We could also have done this by dividing the order of Image may be NSFW.
Clik here to view. $GL_n(\mathbb{F}_p)$ by the order of the Borel subgroup Image may be NSFW.
Clik here to view., but again, doing it this way we learn more, and in fact we get an independent proof of the formula

Image may be NSFW.
Clik here to view. $\displaystyle |GL_n(\mathbb{F}_p)| = p^{ {n \choose 2} } (p - 1)^n [n]_p!$

for the order of Image may be NSFW.
Clik here to view. $GL_n(\mathbb{F}_p)$ , where all three factors acquire clear interpretations: the first factor is the order of the unipotent subgroup Image may be NSFW.
Clik here to view., the second factor is the order of the torus Image may be NSFW.
Clik here to view. $B/U \cong (\mathbb{F}_p^{\times})^n$ , and the third factor is the size of the flag variety Image may be NSFW.
Clik here to view. G/B .

What is going on in these proofs?

Let’s take a step back and compare these explicit proofs of the Sylow theorems for Image may be NSFW.
Clik here to view. $GL_n(\mathbb{F}_p)$ to the general proofs of the Sylow theorems. The first three proofs we gave of Sylow I (reduction to Image may be NSFW.
Clik here to view. S_n , reduction to Image may be NSFW.
Clik here to view. $GL_n(\mathbb{F}_p)$ , action on subsets) all proceed by finding some clever way to get a finite group Image may be NSFW.
Clik here to view. to act on a finite set Image may be NSFW.
Clik here to view. with the following two properties:

Image may be NSFW.
Clik here to view. $|X| \not \equiv 0 \bmod p$ . By the PGFPT this means any Image may be NSFW.
Clik here to view.-subgroups of Image may be NSFW.
Clik here to view. act on Image may be NSFW.
Clik here to view. with fixed points, so we can look for Image may be NSFW.
Clik here to view.-subgroups in the stabilizers Image may be NSFW.
Clik here to view. $\text{Stab}_G(x), x \in X$ .
The stabilizers of the action of Image may be NSFW.
Clik here to view. on Image may be NSFW.
Clik here to view. are Image may be NSFW.
Clik here to view.-groups. Combined with the first property, this means that at least one stabilizer must be a Sylow Image may be NSFW.
Clik here to view.-subgroup, since at least one stabilizer must have index coprime to Image may be NSFW.
Clik here to view..

In fact finding a transitive such action is exactly equivalent to finding a Sylow Image may be NSFW.
Clik here to view.-subgroup Image may be NSFW.
Clik here to view., since it must then be isomorphic to the action of Image may be NSFW.
Clik here to view. on Image may be NSFW.
Clik here to view. G/P . The nice thing, which we make good use of in the proofs, is that we don’t need to restrict our attention to transitive actions, because the condition that Image may be NSFW.
Clik here to view. $|X| \not \equiv 0 \bmod p$ has the pleasant property that if it holds for Image may be NSFW.
Clik here to view. then it must hold for at least one of the orbits of the action of Image may be NSFW.
Clik here to view. on Image may be NSFW.
Clik here to view.. (This is a kind of “Image may be NSFW.
Clik here to view. $\bmod p$ pigeonhole principle.”)

In the explicit proof for Image may be NSFW.
Clik here to view. $G = GL_n(\mathbb{F}_p)$ we find Image may be NSFW.
Clik here to view. by starting with the action of Image may be NSFW.
Clik here to view. on the set of nonzero vectors Image may be NSFW.
Clik here to view. $\mathbb{F}_p^n \setminus \{ 0 \}$ , which satisfies the first condition but not the second, and repeatedly “extending” the action (to pairs of a nonzero vector Image may be NSFW.
Clik here to view. v_1 and a nonzero vector in the quotient Image may be NSFW.
Clik here to view. $\mathbb{F}_p^n/v_1$ , etc.) until we arrived at an action satisfying both conditions, namely the action of Image may be NSFW.
Clik here to view. on the set of tuples of a nonzero vector Image may be NSFW.
Clik here to view. v_1 , a nonzero vector Image may be NSFW.
Clik here to view. $v_2 \in \mathbb{F}_p^n/v_1$ , a nonzero vector Image may be NSFW.
Clik here to view. $v_3 \in \mathbb{F}_p^n / \text{span}(v_1, v_2)$ , etc. (a slightly decorated version of a complete flag).