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 and respectively. The goal of this post is to understand the Sylow -subgroups of in more detail and see what we can learn from them about Sylow subgroups in general.
Explicit Sylow theory for
Our starting point is the following.
Baby Lie-Kolchin: Let be a finite -group acting linearly on a finite-dimensional vector space over . Then fixes a nonzero vector; equivalently, has a trivial subrepresentation.
Proof. If then there are nonzero vectors in , so by the PGFPT fixes at least one of them (in fact at least but these are just given by scalar multiplication).
Now we can argue as follows. If is a finite -group acting on an -dimensional vector space over (equivalently, up to isomorphism, a finite -subgroup of ), it fixes some nonzero vector . Writing , we get a quotient representation on , on which fixes some nonzero vector, which we lift to a vector , necessarily linearly independent from , such that acts upper triangularly on .
Continuing in this way we get a sequence of linearly independent vectors (hence a basis of ) and an increasing sequence of subspaces of that leaves invariant, satisfying the additional condition that fixes . The subspaces form a complete flag in , and writing the elements of as matrices with respect to the basis , we see that the conditions that leaves invariant and fixes 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 is conjugate to a subgroup of the unipotent subgroup .
This is almost a proof of Sylow I and II for (albeit at the prime only), but because we defined Sylow -subgroups to be -subgroups having index coprime to , we’ve only established that is maximal, not that it’s Sylow.
We can show that it’s Sylow by explicitly dividing its order into the order of 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 has no elements of order .
Corollary (Explicit Sylow I and II for : is a Sylow -subgroup of .
Proof. The normalizer is the stabilizer of acting on the set of conjugates of . We want to show that , which would mean that the action of on the conjugates of can be identified with the quotient .
This quotient is the complete flag variety: it can be identified with the action of on the set of complete flags in , since the action on flags is transitive and the stabilizer of the standard flag
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
we can consider the subgroup of which preserves the flag (so ) and which has the additional property that the induced action on 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 of , it has a -dimensional invariant subspace acting on , quotienting by this subspace produces a unique -dimensional invariant subspace acting on , 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 as desired.
(This argument works over any field.)
From here it’s not hard to also prove
Explicit Sylow III for : The number of conjugates of in divides the order of and is congruent to .
Proof. Actually we can compute exactly: we established above that it’s the number of complete flags in (on which acts transitively, hence the divisibility relation), and a classic counting argument (count the number of possibilities for , then for , etc.) gives the -factorial
which is clearly congruent to .
We could also have done this by dividing the order of 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
for the order of , 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 , and the third factor is the size of the flag variety .
What is going on in these proofs?
Let’s take a step back and compare these explicit proofs of the Sylow theorems for to the general proofs of the Sylow theorems. The first three proofs we gave of Sylow I (reduction to , reduction to , 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:
- . By the PGFPT this means any -subgroups of act on with fixed points, so we can look for -subgroups in the stabilizers .
- 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 . 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 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 “ pigeonhole principle.”)
In the explicit proof for we find by starting with the action of on the set of nonzero vectors , which satisfies the first condition but not the second, and repeatedly “extending” the action (to pairs of a nonzero vector and a nonzero vector in the quotient , etc.) until we arrived at an action satisfying both conditions, namely the action of on the set of tuples of a nonzero vector , a nonzero vector , a nonzero vector , 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 ?