## September 19, 2018

### p-Local Group Theory

#### Posted by John Baez I’ve been trying to learn a bit of the theory of finite groups. As you may know, Sylow’s theorems say that if you have a finite group $G$, and $p^k$ is the largest power of a prime $p$ that divides the order of $G$, then $G$ has a subgroup of order $p^k$, which is unique up to conjugation. This is called a Sylow $p$-subgroup of $G$.

Sylow’s theorems also say a lot about how many Sylow $p$-subgroups $G$ has. They also say that any subgroup of $G$ whose order is a power of $p$ is contained in a Sylow $p$-subgroup.

I didn’t like these theorems as an undergrad. The course I took whizzed through them in a desultory way. And I didn’t go after them myself: I was into group theory for its applications to physics, and the detailed structure of finite groups doesn’t look important when you’re first learning physics: what stands out are continuous symmetries, so I was busy studying Lie groups.

Since I didn’t really master Sylow’s theorems, and had no strong motive to do so, I didn’t like them — the usual sad story of youthful mathematical distastes.

But now I’m thinking about Sylow’s theorems again, especially pleased by Robert A. Wilson’s one-paragraph proof of all three of these theorems in his book The Finite Simple Groups. And I started wondering if the importance of groups of prime power order — which we see highlighted in Sylow’s theorems and many other results — is all related to localization in algebraic topology, which is a technique to focus attention on a particular prime.

It’s a nicely written article, so it’s a bit pointless to summarize it, but still, here’s one of the first punchlines.

There’s a functor $X \mapsto \hat{X}_p$ from $Top$ to $Top$, called $p$-completion. A map $f \colon X \to Y$ induces a homotopy equivalence $\hat{f}_p \colon \hat{X}_p \to \hat{Y}_p$ iff it induces an isomorphism of cohomology groups $H^\ast(X,\mathbb{F}_p) \to H^\ast(Y,\mathbb{F}_p)$.

So, $p$-completion takes a space and destroys all information about it that’s not visible using cohomology with $\mathbb{F}_p$ coefficients, while damaging that space as little as possible.

If you hand me a finite group $G$, there’s a strong relation between the $p$-completion of the classifying space $B G$ and the Sylow $p$-subgroups of $G$. The paper explains this in detail.

The rough idea is that one can recover $\widehat{B G}_p$ from a category where:

• the objects are $p$-subgroups of $G$: that is, subgroups having order a power of $p$;

• the morphisms are ways of conjugating one of these $p$-subgroups and then including it in another.

This category is called the fusion category of $G$ at the prime $p$. Since every $p$-subgroup is contained in a Sylow $p$-subgroup and all Sylow $p$-subgroups are conjugation, the fusion category is equivalent to the full subcategory where:

• the objects are subgroups of a fixed Sylow $p$-subgroup of $G$;

• the morphisms are ways of conjugating one of these $p$-subgroups and then including it in another.

Roughly, you can get $\widehat{BG}_p$ by taking the nerve of the fusion category and then $p$-completing it. But this isn’t really true: we need a fancier category, similar to the fusion category, to make something like this true. In this fancier category the objects are not all $p$-subgroups, but only certain special ones called ‘$p$-centric subgroups’, and the hom-sets need to be adjusted a bit too.

There are, however, some results for which the fusion category is enough. For example, Cartan and Eilenberg showed that $H^\ast(B G, \mathbb{F}_p)$ is determined by the fusion category of $G$.

So I don’t yet see what’s really going on, but I’m encouraged that there’s something going on here. I would like, someday, to have a more conceptual — for example, a homotopical — understanding of what Sylow’s theorems are telling us. And I’d also like to understand the analogy between Sylow $p$-subgroups of a finite group and maximal tori in a compact Lie group.

Do you have any clues that can help me? The simpler the better, if possible.

Posted at September 19, 2018 3:52 PM UTC

TrackBack URL for this Entry:   https://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/3060

### Re: p-Local Group Theory

There’s a generalisation of Sylow theory to finite ∞-groups, that is, ∞-groups with finitely many non-trivial homotopy groups which are all finite. See

• Matan Prasma, Tomer Schlank, Sylow theorems for ∞-groups, (arXiv:1602.04494)
Posted by: David Corfield on September 19, 2018 10:37 PM | Permalink | Reply to this

### Re: p-Local Group Theory

Hey, that’s a great paper! Those ‘finite $\infty$-groups’ are precisely those whose $\infty$-groupoid cardinality is well-defined as an alternating product without using any clever tricks for extracting answers from divergent products. And the use of Postnikov towers reminds me a lot of what Mike Shulman and I were discussing in Lectures on n-categories and cohomology.

All the results in this paper could be considered ‘homotopy theory’, and could perhaps have been done a long time ago under that guise, but thinking of pointed connected homotopy types as ‘$\infty$-groups’ makes it a lot more easy to dream up things like this generalization of Sylow’s theorem.

This generalization of Sylow’s theorems is very similar to the original Sylow theorems. That’s nice! But for this very reason, it doesn’t seem to satisfy my thirst for a deeper understanding of what’s ‘really going on’ in Sylow’s theorem. It might, however, serve as a springboard for such an understanding.

Posted by: John Baez on September 20, 2018 2:26 AM | Permalink | Reply to this

### Re: p-Local Group Theory

If it’s only one paragraph, any chance of reproducing Wilson’s proof here?

Posted by: Todd Trimble on September 20, 2018 4:57 AM | Permalink | Reply to this

### Re: p-Local Group Theory

It’s more like a proof sketch than a proof, but I like its brevity:  I posted these as part of a series of tweets when I was trying to test the appetite of that crowd for some real math.

Posted by: John Baez on September 20, 2018 3:04 PM | Permalink | Reply to this

### Re: p-Local Group Theory

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.

Posted by: Mark Meckes on September 20, 2018 5:00 PM | Permalink | Reply to this

### Re: p-Local Group Theory

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.

Posted by: John Baez on September 20, 2018 7:54 PM | Permalink | Reply to this

### Re: p-Local Group Theory

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.

Posted by: Todd Trimble on September 21, 2018 2:04 AM | Permalink | Reply to this

### Re: p-Local Group Theory

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:

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.

Posted by: John Baez on September 23, 2018 1:11 AM | Permalink | Reply to this

### Re: p-Local Group Theory

I don’t have much response yet, although I have some glancing vague familiarity with some of the buzzwords. It does look like something that can be sorted out with a bit of effort, and I agree it does look like there must be a connection between $p$-localizations and the standard proofs. Hmm…

Meanwhile, I’ll just take a moment to acknowledge your observation in slightly expanded form: that an equation equivalent to the class equation I wrote down above,

$\frac{{|A|}}{{|G|}} = \sum_{orbits\; x} \frac1{{|Stab(a_x)|}}\; ,$

is a formula for a groupoid cardinality, in this case of the action groupoid $A//G$ attached to the group action.

Posted by: Todd Trimble on September 23, 2018 1:50 AM | Permalink | Reply to this

### Re: p-Local Group Theory

By the way, this means that in each of your nice applications of the class equation there lurks a groupoid! For example,

• 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.

Here you’re looking at the weak quotient $G//Ad(G)$ of $G$ acting on itself by conjugation… which is actually better than a mere groupoid: it’s a categorical group!

(And right around here is where I kinda regret having called categorical groups “2-groups”. Jim Stasheff warned me of the terminological conflict at the time but I barrelled ahead: now I’m stuck saying things like $G//Ad(G)$ is a “2-$p$-group”, which is particularly embarrassing when $p = 2$.)

It’s not clear if we get much from this viewpoint, but I’m tempted to hope that if I knew more about finite groups I could reinterpret more results using ideas about categorical groups and the like. I suppose homotopy theorists have already done this in their own language.

Posted by: John Baez on September 23, 2018 2:49 AM | Permalink | Reply to this

### Re: p-Local Group Theory

I wanted to jot down a quick thought prompted by Ben’s quick proof: isn’t the $p$-Sylow subgroup of the big group

$H = GL_n(\mathbb{Z}/(p))$

($n$ the order of our group $G$) just a Borel $B$, so that we are letting $G$ act on a flag variety $H/B$?

Posted by: Todd Trimble on September 23, 2018 5:00 PM | Permalink | Reply to this

### Re: p-Local Group Theory

Well, I guess not quite a Borel $B$, but Borel mod a maximal torus $B/T$.

Posted by: Todd Trimble on September 23, 2018 5:03 PM | Permalink | Reply to this

### Re: p-Local Group Theory

I think that it is not quite correct to say that $B/T$ is a Sylow $p$-subgroup of $G$; that’s only a subquotient of $G$, not a subgroup. Probably one should say that the unipotent radical $U$ of the Borel, which does project isomorphically to $B/T$, is a Sylow $p$-subgroup. See, for example, John’s comment or my comment to this effect.

Posted by: L Spice on September 25, 2018 4:18 PM | Permalink | Reply to this

### Re: p-Local Group Theory

Posted by: Todd Trimble on September 25, 2018 5:32 PM | Permalink | Reply to this

### Re: p-Local Group Theory

Let me see if I get it: it looks like for $G = GL(n,k)$, a maximal unipotent $U \subseteq G$ (which is really defined to be a maximal closed connected unipotent subgroup of a linear algebraic group) can be identified with the flag variety $G/B$, where $B \subseteq G$ is a Borel (a maximal closed connected solvable subgroup of a linear algebraic group).

Is this true for all linear algebraic groups? Or maybe under some conditions?

All this would help me set L. Spice’s much earlier comments in proper context:

In a finite group of Lie type—the analog over a finite field of a (usually linear) reductive Lie group—a Sylow $p$-subgroup literally is the unipotent radical of a Borel subgroup.

and

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

which tell me that for a linear algebraic group $G$ over a finite field of characteristic $p$, the maximal unipotents are the same as the $p$-Sylow subgroups.

I would be very happy to visualize these as isomorphic to flag varieties $G/B$, though it’s sort of freaky to reinterpret such flag varieties as groups.

Posted by: John Baez on September 26, 2018 1:35 AM | Permalink | Reply to this

### Re: p-Local Group Theory

a maximal unipotent $U \subseteq G$ (which is really defined to be a maximal closed connected unipotent subgroup of a linear algebraic group) can be identified with the flag variety $G/B$, where $B \subseteq G$ is a Borel (a maximal closed connected solvable subgroup of a linear algebraic group).

Actually this is always impossible (except for $G$ a torus) in any meaningful sense. The flag variety $G/B$ is an irreducible projective variety. I don’t know whether this automatically implies that it has a prime-to-$p$ number of rational points, but it’s true in this generality: the Bruhat decomposition implies that $G/B$ is stratified by affine spaces, each of which has a divisible-by-$p$ number of points except the trivial one (corresponding to the identity coset of $B$). In particular, it can’t be identified in any meaningful way with $U$, which has a composition series with quotients isomorphic to $\mathfrak{gl}_1$, hence is a $p$-subgroup.

What is true is that the unipotent radical $U$ of $B$ embeds naturally as a dense open subset of the flag variety $G/\overline B$ corresponding to an opposite Borel subgroup $\overline B$ to $B$.

Is this true for all linear algebraic groups? Or maybe under some conditions?

It is true at least for (connected) reductive groups that unipotent radicals $U$ of Borel subgroups $B$ of $G$ are Sylow $p$-subgroups of $G$.

Our argument above that $G/B$ has prime-to-$p$ many rational points shows that a Sylow $p$-subgroup of $B$ is one of $G$, and that $U$ is a $p$-group. (Actually one also needs to know something about Galois cohomology to see that $(G/B)(k)$ equals $G(k)/B(k)$; but Galois cohomology of connected groups over finite fields is trivial, so we are fine.) The quotient $B/U$ is isomorphic, in the reductive case, to a maximal torus $T$ of $G$, whose number of rational points is prime to $p$ (for example, because every element is semisimple, hence has prime-to-$p$ order).

Posted by: L Spice on September 26, 2018 9:16 PM | Permalink | Reply to this

### Re: p-Local Group Theory

I don’t know about the analogy between Sylow $p$-subgroups and maximal tori in compact groups, but I know a wrong argument for why there isn’t such an analogy. (That there is one is evidenced, probably, by the existence of the theory of $p$-compact groups, which is one of those things that I really want to but don’t understand.)

In a finite group of Lie type—the analogy over a finite field of a (usually linear) reductive Lie group—a Sylow $p$-subgroup literally is the unipotent radical of a Borel subgroup. Therefore, one might view unipotent radicals of Borel subgroups as the correct analogue for reductive algebraic groups of Sylow $p$-subgroups. Now there is an analogy (at least an informal one, and probably people know how to make it formal) between maximal tori in compact Lie groups and Borel subgroups of quasisplit Lie groups, suggesting that the correct analogue for compact Lie groups of a Sylow $p$-subgroup is the unipotent radical of a maximal torus—but the unipotent radical of a maximal torus is trivial.

Hopefully this much ignorance in one place will prompt someone else to come along and remedy it.

Posted by: L Spice on September 20, 2018 8:08 PM | Permalink | Reply to this

### Re: p-Local Group Theory

There’s an MO answer which claims that

The Sylow theorems are finite group analogues of a bunch of results about “maximal unipotent subgroups” in algebraic groups.

Posted by: David Corfield on September 20, 2018 10:22 PM | Permalink | Reply to this

### Re: p-Local Group Theory

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

Posted by: L Spice on September 20, 2018 10:41 PM | Permalink | Reply to this

### Re: p-Local Group Theory

Cool! I’m perfectly happy for Sylow subgroups to be analogues of maximal unipotents rather than maximal tori. The theory of simple Lie groups, simple algebraic groups, and their Lie algebras prominently features maximal subgroups and subalgebras of various kinds, all interrelated, and I didn’t have any deep reason to guess that Sylow subgroups were like maximal tori instead of one of these other things.

Posted by: John Baez on September 20, 2018 10:30 PM | Permalink | Reply to this

### Re: p-Local Group Theory

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

Posted by: L Spice on September 26, 2018 7:50 PM | Permalink | Reply to this

### Re: p-Local Group Theory

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.

Posted by: John Baez on September 26, 2018 8:51 PM | Permalink | Reply to this

### Re: p-Local Group Theory

Maybe it’s easier to see what the relationship is between the $p$ Sylow subgroups and the pro $p$ completion of groups, which can be defined as here

https://ncatlab.org/nlab/show/profinite+completion+of+a+group. Actually this might not be so exciting: the pro $p$ completion is given by first taking the nilpotent completion as the limit of the group modulo its central series: $G/[G, G]_n$, and then breaking up into prime factors.

So for a simple group, the $p$ completion is just $e$, in contrast to its homotopy $p$ completion which is a space with all sorts of interesting homology.

I guess the lossiness of the ordinary completion reflects the fact that we can only recover nilpotent spaces by looking locally at every prime. Maybe the $p$ Sylow subgroups capture more subtle $p$-ary information than the completions do.

For rational homotopy types it is possible to remember more information about spaces with non-nilpotent fundamental groups by considering stacks rather than just commutative dg algebras. Is something similar possible p adically?

Posted by: Phil Tosteson on September 21, 2018 6:01 PM | Permalink | Reply to this

### Re: p-Local Group Theory

My favorite proof of Sylow is the following.

First observe if $G$ has a $p$-Sylow $P$ so does all its subgroups. (Have the subgroup act on $G/P$ and note the stabilizers are $p$-groups and some orbit has order prime to $p$).

Note $GL_n(Z/p)$ has a p-Sylow subgroup: the unitriangular matrices.

$S_n$ embeds in $GL_n$ as permutation matrices.

All groups embed in $S_n$.

Posted by: Benjamin Steinberg on September 22, 2018 12:25 PM | Permalink | Reply to this

### Re: p-Local Group Theory

Posted by: Todd Trimble on September 22, 2018 1:06 PM | Permalink | Reply to this

### Re: p-Local Group Theory

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

Posted by: David Corfield on September 22, 2018 4:28 PM | Permalink | Reply to this

### Re: p-Local Group Theory

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?

Posted by: David Corfield on September 22, 2018 6:11 PM | Permalink | Reply to this

### Re: p-Local Group Theory

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 “$\mathbb{F}_{1^n} = \mathbb{F}$”.

Posted by: John Baez on September 22, 2018 9:57 PM | Permalink | Reply to this

### Re: p-Local Group Theory

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 $p$.

Posted by: David Corfield on September 23, 2018 9:09 AM | Permalink | Reply to this

### Re: p-Local Group Theory

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.

Posted by: David Corfield on September 23, 2018 10:07 AM | Permalink | Reply to this

### Re: p-Local Group Theory

Having written up Ben’s slick proof in the nLab article on Sylow subgroups, it occurs to me that the related result Theorem 3.4 on the class equation page, that there is a subgroup of order $p^i$ for any $p^i$ dividing the order of $G$, is a trivial consequence of the maximal Sylow case. Just use the already established solvability of $p$-groups to infer a composition series of a $p$-Sylow group $P$

$\{1\} = P_0 \subset P_1 \subset \ldots \subset P$

where each $P_i$ has order $p^i$. This seems much cleaner than the proof of 3.4 adapted from the Wikipedia proof, which involves some arithmetic on binomial coefficients that I find slightly grubby (as well as some slightly fancy footwork in the opening paragraph, which might be hard for others to follow).

I’m a little conflicted as to whether to let the grubby proof stand, or dismantle it and replace it. Maybe I’ll archive the grubby proof at the nForum and link to it for posterity, and replace it.

Posted by: Todd Trimble on September 23, 2018 3:32 PM | Permalink | Reply to this

### Re: p-Local Group Theory

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:

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.

Posted by: John Baez on September 23, 2018 7:06 PM | Permalink | Reply to this

### Re: p-Local Group Theory

Thanks! You taught me something I didn’t know before (pretty easy to do, admittedly, when it comes to group theory).

Posted by: Todd Trimble on September 23, 2018 7:37 PM | Permalink | Reply to this

### Re: p-Local Group Theory

…a finite nilpotent group is a direct product of its Sylow p–subgroups.

Perhaps some other useful things there.

Posted by: David Corfield on September 23, 2018 9:09 PM | Permalink | Reply to this

### Re: p-Local Group Theory

David wrote:

…a finite nilpotent group is a direct product of its Sylow $p$–subgroups.

Excellent! You seem to have read my mind: I was just wondering to what extent we could recover a finite group from its Sylow $p$-subgroup.

For example, if we have an element $g$ of order 6 in a finite group, does is lie in the subgroup generated by some Sylow 2-subgroup and some Sylow 3-subgroup? Yes, because $g^3$ is of order 2 so lies in a Sylow 2-subgroup, and similarly $g^2$ lies in a Sylow 3-subgroup. This seems to generalize to arbitrary products of primes using the Chinese remainder theorem.

But this is a very crude statement since I needed to choose my Sylow 2-subgroup and Sylow 3-subgroup to get this to work, and I have little insight into this choice or how the Sylow $p$-subgroups interact to build up the whole group.

But the case of nilpotent group is heavenly! The group just breaks neatly into a product of its Sylow $p$-subgroups, which can be arbitrary $p$-groups.

And this give me more of a feeling for why $p$-localization is better for nilpotent spaces, meaning spaces with nilpotent $\pi_1$ acting nilpotently on the higher homotopy groups. I still don’t see exactly what ‘acting nilpotently on the higher homotopy groups’ buys you, but now we see $\pi_1$ being nilpotent means it breaks up into pieces, one for each prime.

By the way, I’m guessing the proof of the result you mentioned goes like this: if $G$ is a $p$-group and $A$ is an abelian $q$-group with $q$ some prime distinct from $p$, then $H^2(G,A) = 0$. This prevents us from getting central extensions of $G$ by $A$ that aren’t just direct products.

I don’t know how you show $H^2(G,A) = 0$ in this case but it could be just fiddling around with formulas. And I’ll go out on a limb and guess $H^n(G,A) = 0$ under the same assumptions. This would help us take a nilpotent space and break it into $p$-local pieces.

In defining these cohomology groups we really get to choose an action of $G$ on $A$, and it’s possible my guesses are only true when the action of $G$ on $A$ is nilpotent. That would make me sad (because we get a weaker result) but also happy (because then I’d understand why people insert this kind of condition in the definition of nilpotent space).

Studying finite groups and homotopy theory as a hobby is a bit like trying to climb Mt. Everest on weekends. One certainly won’t get very far up, but at least there are nice views sometime.

Posted by: John Baez on September 23, 2018 9:59 PM | Permalink | Reply to this

### Re: p-Local Group Theory

Okay, one thing I need to read is this:

It has nice section titles like “Nilpotent spaces and Postnikov towers”, “Localization of nilpotent groups and spaces”, “Global to local: abelian and nilpotent groups”, and “Local to global: abelian and nilpotent groups”. So it should answer most of the puzzles I was wondering about in my last comment.

Posted by: John Baez on September 24, 2018 10:44 PM | Permalink | Reply to this

### Re: p-Local Group Theory

Here is a nice way to see the Wikistyle proof without messy counting. I’ll only do getting the Sylow subgroup since an easy induction then gives $p$-power subgroup of all size.

Claim. Let $p$ be a prime and $n=p^k\cdot m$. Then $\binom{n}{p^k}\equiv m\mod p$.

Proof. Let $X$ be the set of $p^k$-element subsets of $Z_n$. Let $H=\langle m\rangle$ be the subgroup of $Z_n$ of order $p^k$ (generated by the class of $m$). Then $H$ acts on $X$ by $h+A=\{h+a\mid a\in A\}$ for $A\in X$. Since $H$ is a $p$-group, we get that $|X|\equiv |X_f|\mod p$ where $X_f$ is the set of fixed points. Now $|X|=\binom{n}{p^k}$. We claim that $X_f$ is the set $Z_n/H$ of cosets of $H$. Since $[Z_n:H]=m$, it will then follow that $\binom{n}{p^k}\equiv m\mod p$. Indeed, a subset of $Z_n$ is fixed by $H$ if and only if it is a union of cosets, so a subset of size $p^k$ fixed by $H$ is a single coset. This establishes the claim.

From this it follows that a group $G$ of order $n$ (as above). Just have $G$ act on its $p^k$-element subsets via translation. Since the number of such subsets is prime to $p$ and the stabilizers are of size a $p$-power we are done

Posted by: Benjamin Steinberg on September 24, 2018 3:44 PM | Permalink | Reply to this

Post a New Comment