**EDIT 21/12:** Even if there are no conclusive answers to these questions, I would very much like to know if anyone has **noted** and **attempted to explain** the mysterious significance of local subgroups: are there any papers/books that offer some insights about this?

Let $G$ be a finite group, and let $p$ be a prime. A *$p$-local subgroup* of $G$ is the normalizer of some $p$-subgroup of $G$.

**The mystery:** Strangely, the local subgroups of a finite group tell us a lot about the structure of the group. There are many **local-to-global** results, which extract information about a finite group given information about its local subgroups.

## Evidence

**Theorem:** (Brauer-Fowler) Let $G$ be a finite simple group. Let $t\in G$ be an involution, and suppose $|\mathbf{C}_G(t)|=n$. Then $|G|<(n^2)!$.

Notice that the centralizer of an involution is a $2$-local subgroup. This theorem tells us that there are at most finitely-many finite simple groups with a given centralizer-of-involution. In fact, in some instances, given a group, we can classify all finite simple groups with an involution whose centralizer is isomorphic to that given group.

**Theorem:** Let $G$ be a finite simple group. Suppose that $G$ has an involution $t$, such that $\mathbf{C}_G(t)\cong D_8$, where $D_8$ is the dihedral group with $8$ elements. Then $G\cong A_6\text{ or }PSL(3,2)$.

**Theorem:** Let $G$ be a finite simple group. Suppose that $G$ has an involution $t$, such that $\mathbf{C}_G(t)\cong C_2\times C_2$. Then $G\cong A_5$.

Let $p$ be a prime. A **normal $p$-complement** of a finite group $G$, is a normal subgroup $N$ such that $|G:N|$ is a power of $p$ and $|N|$ is not divisible by $p$. In other words, $|G:N|$ is the largest power of $p$ dividing $|G|$.

**Theorem:** (Frobenius) Let $G$ be a finite group, and fix a prime $p$. If every normalizer of a non-trivial $p$-subgroup of $G$ has a normal $p$-complement, then $G$ itself has a normal $p$-complement.

**Theorem:** Let $G$ be a finite group, and suppose that for every odd prime $p$, every $p$-local subgroup of $G$ has a normal Sylow $2$-subgroup. Then $G$ itself has a normal Sylow $2$-subgroup.

**Brauer's Main Theorems** in modular representation theory relate the blocks of a finite group in characteristic $p$ with those of its $p$-local subgroups. I admit I don't know anything about modular representation theory, but it's interesting that local subgroups enter modular representation theory in a natural way. Perhaps this could be a route towards understanding the mysterious significance of local subgroups in finite group theory.

Here is an open conjecture that has been verified in many special cases:

**McKay Conjecture:** Let $G$ be a finite group. Define $\text{Irr}_{p'}(G)$ to be the set of irreducible complex characters of $G$ whose degree is **not** divisible by $p$. Then for any Sylow $p$-subgroup $P\subseteq G$, $|\text{Irr}_{p'}(G)|=|\text{Irr}_{p'}(\mathbf{N}_G(P))|$.

(This conjecture has been proven in the case $p=2$.)

*Edit 22/12:* (contributed by @Geoff Robinson in the comments) The Thompson Order Formula says that if a finite group $G$ has more than one conjugacy class of involutions, then the order of $G$ is determined by $2$-local data.

My question is: **Is there some satisfying explanation for this mysterious significance of local subgroups?**

I don't know what a satisfying answer to this question would look like. I vaguely recall reading (and would very much appreciate it if someone could provide a reference) that local group-theoretic analysis is somehow analogous to Tits' theory of buildings. Has anyone elaborated on this analogy?

I read this in Ronald Solomon's article, *"A Brief History of the Classification of the Finite Simple Groups"*:

*Indeed the local subgroups afford the largest
proper subgroups of G whose existence can be predicted a priori from knowledge
of $|G|$ alone.*

How can this be made precise?

simplegroup $\endgroup$1more comment