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 p-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 P be a finite p-group acting linearly on a finite-dimensional vector space V over \mathbb{F}_p. Then P fixes a nonzero vector; equivalently, V 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 P 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 P is a finite p-group acting on an n-dimensional vector space V over \mathbb{F}_p (equivalently, up to isomorphism, a finite p-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 P fixes some nonzero vector, which we lift to a vector v_2 \in V, necessarily linearly independent from v_1, such that P 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 V) and an increasing sequence V_k = \text{span}(v_1, \dots v_k) of subspaces of V that P leaves invariant, satisfying the additional condition that P fixes v_k \bmod V_{k-1}. The subspaces V_k form a complete flag in V, and writing the elements of P as matrices with respect to the basis v_1, \dots v_n, we see that the conditions that P leaves V_k invariant and fixes v_k \bmod V_{k-1} says exactly that P acts by upper triangular matrices with 1s on the diagonal in this basis.

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

Proposition: Every p-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 p only), but because we defined Sylow p-subgroups to be p-subgroups having index coprime to p, 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 p-subgroup P of a group G is Sylow iff the quotient N_G(P)/P has no elements of order p.

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 p.

Corollary (Explicit Sylow I and II for GL_n(\mathbb{F}_p): U_n(\mathbb{F}_p) is a Sylow p-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 U. We want to show that N_G(U) = B, which would mean that the action of G on the conjugates of U can be identified with the quotient G/B.

This quotient is the complete flag variety: it can be identified with the action of G 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 B. So it suffices to exhibit a G-equivariant bijection between conjugates of U and complete flags which sends U 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 i. This produces U when applied to the standard flag, so produces conjugates of U when applied to all flags. In the other direction, given a conjugate gUg^{-1} of U, it has a 1-dimensional invariant subspace V_1 acting on \mathbb{F}_p^n, quotienting by this subspace produces a unique 1-dimensional invariant subspace V_2/V_1 acting on \mathbb{F}_p^n/V_1, etc.; this produces the standard flag when applied to U, so produces g applied to the standard flag when applied to conjugates of U. So we get our desired G-equivariant bijection between conjugates of U 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 p-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 B, 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 U, 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 G to act on a finite set X with the following two properties:

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

In fact finding a transitive such action is exactly equivalent to finding a Sylow p-subgroup P, since it must then be isomorphic to the action of G 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 X then it must hold for at least one of the orbits of the action of G on X. (This is a kind of “\bmod p pigeonhole principle.”)

In the explicit proof for G = GL_n(\mathbb{F}_p) we find X by starting with the action of G 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 G 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).

The next question we’ll address in Part III is: can we do something similar for G = S_n?