*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.

The answer is yes:

- Carles Broto, Ran Levi, and Bob Oliver. The theory of $p$-local groups: a survey,
*Contemporary Mathematics***346**(2004), 51–84.

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.

## 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

Sylow theorems for ∞-groups, (arXiv:1602.04494)