The n-Category Café

That looks like (an indication of) what I think of as the standard proofs.

I like the brevity, too. This is a painting in broad strokes of a forest, which could easily be obscured by describing the trees in detail.

Right: for mathematicians of a certain level of experience, packing a proof with too many details can be more distracting than helpful. It’s amusing to compare Wilson’s argument to the proof on Wikipedia, which features horrors like

$$|\Omega | =\binom{p^k m}{p^k} = \prod_{j=0}^{p^k - 1} \frac{p^k m - j}{p^k - j} = m\prod_{j=1}^{p^{k} - 1} \frac{p^{k - \nu_p(j)} m - j/p^{\nu_p(j)}}{p^{k - \nu_p(j)} - j/p^{\nu_p(j)}}$$

This is the sort of thing I’d rather come up with myself, in my own notation, to check one of Wilson’s claims: namely, if $p^k$ is the largest power of $p$ dividing the order of $G$, then

$$\binom{|G|}{p^k}$$

is not divisible by $p$. I am happy that Wilson trusts me to check this claim myself if I want, or skip checking it if I don’t.

Cool; thanks. It’s nice.

I had the same general feeling as you when first encountering the Sylow theorems and such: they didn’t excite my interest and I found all of it unmemorable.

A while back I went over it again and my current understanding is that these are among the sundry results that boil down to applications of the class equation for group actions. Maybe it wasn’t clear to me way back when that it wasn’t just about groups, but group actions.

The class equation in its categorified form is almost a tautology: it says that if $G$ acts on a set $A$, then there is an isomorphism of $G$-sets

$$A \cong \sum_{orbits\; x} G/Stab(a_x)$$

where a representative element $a_x$ is chosen for each orbit $x$, and $Stab(a_x)$ denotes its stabilizer subgroup. The sum is just like how we break up a locally connected space as a coproduct of connected components, and indeed the topos of $G$-sets is a locally connected topos: every object $A$ is a sum of connected components (= transitive $G$-sets) according to the above isomorphism.

Taking cardinalities,

$${|A|} = \sum_{orbits\; x} \frac{{|G|}}{{|Stab(a_x)|}}$$

which is how the class equation is usually expressed in decategorified form.

The class equation is used to derive all of the following results (as shown in the nLab):

• Every nontrivial $p$-group $G$ has a nontrivial center (hence $G$ is solvable). For this one considers the action of $G$ on itself by conjugation.

• If a $p$-group $G$ acts on a set $A$, then the number of fixed points of $A$ is congruent to ${|A|}$ modulo $p$.

• Wedderburn’s theorem: every finite division ring $D$ is commutative. Sketch: put $F = Z(D)$. If $\dim_F D = n$ and ${|F|} = q$, then the conjugation action of $D^\times$ on itself leads to a class equation of type $$q^n - 1 = q - 1 + \sum_{orbits\; x} \frac{q^n - 1}{q^{d_x} - 1}$$ where $d^x$ is the dimension over $F$ of the centralizer of an orbit representative $a_x$. One can then deduce that the product of all $q - \zeta$, where $\zeta$ ranges over primitive $n^{th}$ roots of unity, divides $q-1$, an absurdity unless $n=1$.

• If $p^k$ divides ${|G|}$, then the group $G$ has a subgroup of order $p^k$ (in particular, $p$-Sylow subgroups, where $p^k$ is the largest possible $p$-power divisor, exist). One considers how $G$ acts by left translation on $\binom{G}{p^k}$, the set of subsets of $G$ of cardinality $p^k$. The proof in the nLab is adapted from the Wikipedia proof; only John can say whether he likes it any better :-).

• Any $p$-subgroup $H$ is conjugate to a subgroup of any $p$-Sylow subgroup $P$ (hence any two $p$-Sylow subgroups are conjugate). For this one considers the action of $G$ on $G/P$ and restricts to an action of $H$ on $G/P$. Since ${|G/P|}$ is prime to $p$, so is ${|Fix_H(G/P)|}$ according to the second bullet point. In particular ${|Fix_H(G/P)|}$ is nonempty; if $g P$ is fixed under $H$, i.e., $h g P = g P$ for all $h \in H$, then $g^{-1} h g \in P$ for all $h \in H$.

• The number of $p$-Sylow subgroups is congruent to $1$ modulo $p$. For this one considers the action of $G$ on $p$-Sylow subgroups by conjugation, using the last bullet point.

As indicated, in each case the proof begins by choosing the right group action.

Thanks, Todd! I only recently learned the term “class equation”, though I knew the fact. It makes a nice partner with Burnside’s Lemma — so called because it’s due to Frobenius. I like comparing these formulas to the simpler one you get using groupoid cardinality: if $G$ is a finite group acting on a finite set $A$, then

$$|A//G| = |A|/|G| .$$

While Sylow’s theorem’s are “among the sundry results that boil down to applications of the class equation for group actions”, I still feel they’re pointing to something that I can’t quite put my finger on. Something about how a thing (here a finite group, but also a homotopy type or a scheme) can be viewed “through the lens of any prime”, and these views for various primes add up to give us a pretty good, though often not complete, picture of the original thing. This theme comes up over and over in algebra, number theory and algebraic topology, and there are pretty general ways of making this theme precise:

• nLab, p-localization of a space.

• nLab, fracture theorem.

• Wikipedia, Hasse principle.

However, I don’t see how (or even whether) Sylow’s theorems fit into this theme. My blog post represents my best attempt so far.

Yup! In the finite-field setting (where all groups are quasisplit), a maximal (connected) unipotent subgroup is a unipotent radical of a Borel subgroup.

Incidentally, Richard Montgomery believed in 2014 on MathOverflow that the analogy you suggest (between Sylow $p$-subgroups and maximal tori) is appropriate.

That’s interesting! But his reasoning was just as loose as mine: “because any two are conjugate”.

I think you’ve sharpened the analogy using maximal unipotents instead of maximal tori.

I guess you could go through this argument imagining $p =1$. So is then a 1-Sylow subgroup just the group itself, with $S_n$ the maximal unipotent subgroup of itself?

David wrote:

Is there a ‘field with one element’ slant possible here with $S_n = GL_n(\mathbb{F}_1)$?

This is one of the things that has me confused.

There’s a sense in which the study of finite groups is the study of algebraic groups over the field with one element: various facts about algebraic groups over $\mathbb{F}_p$ degenerate to facts about finite groups as $p \to 1$.

But then any finite group has Sylow $p$-subgroups for all primes $p$, any one of which is is somehow analogous to a maximal unipotent subgroup of an algebraic group.

So primes seem to be showing up in two different ways here. Or are they the same somehow?

The MathOverflow comment that you mentioned, by Vipul Naik, has a clue at the end:

For a finite field $\mathbb{F}_q$ where $q$ is a power of $p$, the maximal unipotent subgroup of $GL_n(\mathbb{F}_q)$ is the $p$-Sylow subgroup.

It’s perhaps worth adding that Kapranov and Smirnov have thoughts on what algebra over the finite field $\mathbb{F}_q$ with $q = p^n$ degenerates to as $p \to 1$. We don’t get “$\mathbb{F}_{1^n} = \mathbb{F}$”.

But then any finite group has Sylow $p$-subgroups for all primes $p$, any one of which is is somehow analogous to a maximal unipotent subgroup of an algebraic group.

Couldn’t this be seen as reflecting morphisms between fields? Standardly, the only homomorphisms between $\mathbb{F}_m$ and $\mathbb{F}_n$ are injections when $m|n$. The elusive $\mathbb{F}_1$ would inject into all fields, and so its algebraic groups (i.e., finite groups) inherit via base change $p$-Sylows for every $p$.

I guess the “maximal unipotent subgroup” of $S_n$, and hence all finite groups, is the trivial subgroup. The only permutation matrix with all eigenvalues 1 is the identity matrix.

It looks like you updated the proof on the nLab. I think that’s good!

Since I happen to be curious about $p$-groups right now, I added a remark to class equation saying that $p$-groups are actually nilpotent, not just solvable. Let me explain why I did.

I’m fond of the Postnikov tower interpretation for group cohomology, where we think of a cohomology class of degree $n-1$ as instructions for adding a new ‘top layer’ to a homotopy $n$-type and getting a homotopy $(n+1)$-type.

The 2nd cohomology $H^2(G,A)$ is a degenerate case because it classifies ways of starting with a pointed connected homotopy 1-type (the group $G$) and centrally extending it by $A$ to get another pointed connected homotopy 1-type (that is, another group). Here the new ‘top layer’ is at the same level as the original layer.

Now I’m finding it interesting that $p$-groups are all nilpotent, and thus built from iterated central extensions where we take $A = \mathbb{Z}/p$ each time. This is a shocking conceptual simplification compared to full-fledged finite group theory. It’s only a “conceptual” simplification, because in practice classifying $p$-groups is still intractable, due to the wild profusion of these central extensions. But still, it’s interesting!

The paper David pointed out:

• Matan Prasma and Tomer Schlank, Sylow theorems for $\infty$-groups.

generalizes Sylow’s theorems to ‘finite $\infty$-groups’: pointed connected homotopy types with finitely many nontrivial homotopy groups, all of which are finite.

This makes me want to think about the case where all these homotopy groups are $p$-groups, and how the Postnikov tower simplifies in this case. But I guess I’m following very slowly in the footsteps of algebraic topologists who have studied the hell out of $p$-local homotopy types.