The mysterious significance of local subgroups in finite group theory

Question

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?

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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.

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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?

Answer 1

There is indeed a strong analogy between the study of $p$-local subgroups and the theory of buildings, at least for groups of Lie type.

More precisely, if $G$ is a finite group of Lie type over a field of characteristic $p$, then the parabolic subgroups of $G$ are, in fact, $p$-local subgroups. The notion of parabolic subgroups comes from the theory of algebraic groups. There are several ways to define what a parabolic subgroup is. The most "geometric" interpretation is that a closed subgroup $P \leq G$ (closed w.r.t. the Zariski topology) is parabolic if and only if $G/P$ is a projective variety. Another way to define it is that a closed subgroup $P \leq G$ is parabolic if and only if it contains a Borel subgroup, i.e., a maximal closed (smooth) connected solvable subgroup of $G$. In the finite case, these Borel subgroups are precisely the normalizers of the full Sylow $p$-subgroups.

In any case, the building associated with an algebraic group $G$ consist precisely of the collection of all parabolic subgroups ordered by opposite inclusion; this gives rise to a simplicial complex, and this is the building. (This is the point of view taken in Tits' original lecture notes from 1974. There exist other equivalent interpretation of these buildings, e.g. as labelled graphs [chamber systems], but this is perhaps not so relevant for your question.)

A possible reference is the (introduction of) the recent paper "Rank one isolated p-minimal subgroups in finite groups" (Ulrich Meierfrankenfeld, Chris Parker, Peter Rowley, Journal of Algebra, Volume 566, 2021, 1-93), which, in turn, contains a number of other references where more details can be found.

Many of these ideas of associating "geometries" (such as the buildings) to parabolic subgroups go beyond the case of groups of Lie type. See, for instance, the paper "Minimal Parabolic Geometries for the Sporadic Groups" by Mark Ronan and Gernot Stroth (European Journal of Combinatorics 5, 1984, 59-91).

Answer 2

I think this question is vast, and that there is no single answer.

My first remark might be that to produce ANY proper non-cyclic subgroups of a finite group $G$, we have to allow $G$ so act on some structure, and then consider elements of $G$ which preserve some substructure. If you know nothing about an abstractly defined group, where are you going to find structures for it to act on? Firstly, you can consider the normalizers and centralizers of cyclic subgroups of $G$, which are subgroups obtained by considering the conjugation action of $G$ on itself. This leads to the class equation, which is one route to the existence of Sylow $p$-subgroups of $G$ for each prime $p$, and, more generally, the existence of subgroups of order $p^{d}$ of $G$ for each power $p^{d}$ of $p$ which divides $|G|$. Then one may consider normalizers of these $p$-subgroups, the so-called $p$-local subgroups, and how they intersect with each other. so, personally speaking, I do not find it so mysterious that $p$-local analysis should become a powerful tool in finite group theory.

However, I would highlight two general themes which have been extensively studied in the last few decades, both of which are somewhat related to the examples you provide.

One key general theme is that "fusion is local", probably first formulated by J.L. Alperin around 1967, with the celebrated Alperin's fusion theorem. This states that the conjugation within $G$ of the subgroups of a Sylow $p$-subgroup of $G$ can be effect step-by-step within normalizers of certain non-trivial subgroups of $P$ in a very precise fashion. This may be regarded as a vast generalization of a theorem of Burnside in the case that $P$ is Abelian. This married well with theorems on transfer and normal $p$-complements, such as the work of Glauberman and Thompson, which dramatically advanced in the 1960s and 70s.

In more recent years, Alperin's fusion theorem has been formalized and abstracted in other contexts (such as modular representation theory and algebraic topology) and abstracted to the theory of fusion systems, currently a burgeoning topic.

Another theme, which may be viewed as an attempt to generalize properties of Tits buildings to general finite groups, is that of the simplicial complex associated to the poset of non-trivial $p$-subgroups of a finite group $G$ (and various $G$-homotopy equivalent related complexes). This was studied in the 70s by K.S. Brown and D. Quillen, and later by authors such as S. Bouc, J. Thevenaz and P. Webb.

If $G$ is a finite group of order divisible by the prime $p$, we may consider the simplicial complex $\mathcal{S} = \mathcal{S}(G,p)$ associated to the poset of non-trivial $p$-subgroups of $G$, on which $G$ acts by conjugation. We consider $G$ to be the stabilizer of the empty simplex, and for a non-empty $n$-simplex $\sigma = (P_{0} < P_{2} < \ldots < P_{n}),$ (where the $P_{i}$ are non-trivial $p$-subgroups of $G$, each strictly contained in its successor), the chain stabilizer $G_{\sigma}$ is equal to $\bigcap_{i=0}^{n}N_{G}(P_{i})$.

When $A$ is an Abelian group and we have a function $f$ ( depending only on isomorphism type) from $\{$ finite groups$ \} \to A,$ the question "is the function $f$ $p$-locally controlled in a precise fashion?" can often be reduced to the question "is it true that $\sum_{\mathcal{S}/G}(-1)^{|\sigma|} f(G_{\sigma}) = 0$ (for every finite group $G$ of order divisible by $p$)?".

Notice that if the alternating sum is zero, then it is always the case that $f(G)$ is a specific $\mathbb{Z}$-combination of $f(G_{\sigma})'s,$ where $\sigma$ runs over $G$-orbits of non-empty simplices. More subtly (and often overlooked) is the fact that the orbit structure of $\mathcal{S}$ is determined $p$-locally by Alperin's fusion theorem.

Two situation where this occurs are in Webb's theorem on control of cohomology ( with trivial coefficients), where the formula $H^{n}(G,M) = \sum_{\emptyset \neq \sigma/G}(-1)^{|\sigma|+1}H^{n}(G_{\sigma},M)$ is obtained, and in the 1989 work of R. Knoerr and myself on Alperin's weight conjecture, where we prove that Alperin's weight conjecture is true for all finite groups if and only if whenever $B$ is a $p$-block of positive defect of a finite group $G$, and $k(B)$ is the number of irreducible characters in $B$,etc., then we have $\sum_{\sigma \in \mathcal{S}/G} (-1)^{|\sigma|+1}k(B_{\sigma}) = 0,$ where $B_{\sigma}$ is the set of Brauer correspondent blocks of $G_{\sigma}$ associated to $B$ via the Brauer homomorphism.

There are many other instances where such formulae arise naturally in conjectures in modular representation theory, and many questions remain about the methodology of proof required to establish ( or, I suppose, disprove) conjectural formulae of this nature.

I could write much more, but this answer is long enough.

Later edit: As @SteveD mentions, a famous theorem of G.Mislin states that knowledge of the cohomology ring $H^{\ast}(G, \mathbb{F}_{p})$ determines the $p$-fusion in $G$.