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 S_n and $GL_n(\mathbb{F}_p)$ respectively. The goal of this post is to understand the Sylow -subgroups of $GL_n(\mathbb{F}_p)$ in more detail and see what we can learn from them about Sylow subgroups in general.

Explicit Sylow theory for $GL_n(\mathbb{F}_p)$

Our starting point is the following.

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

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

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

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

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

Proposition: Every -subgroup of $GL_n(\mathbb{F}_p)$ is conjugate to a subgroup of the unipotent subgroup $U_n(\mathbb{F}_p)$ .

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

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

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

Corollary (Explicit Sylow I and II for $GL_n(\mathbb{F}_p)$ : $U_n(\mathbb{F}_p)$ is a Sylow -subgroup of $GL_n(\mathbb{F}_p)$ .

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

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

$\displaystyle 0 \subset \text{span}(e_1) \subset \text{span}(e_1, e_2) \subset \dots \subset \mathbb{F}_p^n$

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

But this is clear: given a complete flag

$\displaystyle 0 = V_0 \subset V_1 \subset \dots \subset V_n = \mathbb{F}_p^n$

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

(This argument works over any field.)

From here it’s not hard to also prove

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

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

$\displaystyle n_p = [n]_p! = \prod_{i=1}^n [i]_p = \prod_{i=1}^n \left( p^{i-1} + p^{i-2} + \dots + 1 \right)$

which is clearly congruent to $1 \bmod p$ . $\Box$

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

$\displaystyle |GL_n(\mathbb{F}_p)| = p^{ {n \choose 2} } (p - 1)^n [n]_p!$

for the order of $GL_n(\mathbb{F}_p)$ , where all three factors acquire clear interpretations: the first factor is the order of the unipotent subgroup , the second factor is the order of the torus $B/U \cong (\mathbb{F}_p^{\times})^n$ , and the third factor is the size of the flag variety 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 $GL_n(\mathbb{F}_p)$ to the general proofs of the Sylow theorems. The first three proofs we gave of Sylow I (reduction to S_n , reduction to $GL_n(\mathbb{F}_p)$ , action on subsets) all proceed by finding some clever way to get a finite group to act on a finite set with the following two properties:

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

In fact finding a transitive such action is exactly equivalent to finding a Sylow -subgroup , since it must then be isomorphic to the action of on 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 $|X| \not \equiv 0 \bmod p$ has the pleasant property that if it holds for then it must hold for at least one of the orbits of the action of on . (This is a kind of “ $\bmod p$ pigeonhole principle.”)

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