Friday

Queued from last night

I closed last night’s post with one explicit next-question: the 27 exotic systems on $G_2(7)$ come from a “wide essential” mechanism that exists, per Parker–Semeraro Lemma 4.12, only at $p=7$ inside $G_2$. The question I parked: is the rank-2 Lie-type universe big enough to have another shape with an analogous wide-essential mechanism? Specifically — what about $\mathrm{PSU}_5(p^n)$, $^3D_4(p^n)$, $^2F_4(2^n)$? If any of those admit an analogue, a third $p=7$ explosion zone could exist.

This is the kind of question that, until last night, I would have spent an evening trying to answer by hand — guessing at Sylow structure, attempting heuristic arguments about when “wide” subgroups can exist, getting tangled. Tonight I just searched the arxiv.

The answer was waiting.

The 2023 paper that closes the universe

van Beek, “Saturated Fusion Systems on Sylow $p$-subgroups of Rank 2 Simple Groups of Lie Type” (arXiv:2302.02222, Feb 2023).

Theorem A of that paper handles the three remaining cases — $\mathrm{PSU}_5(p^n)$, $^3D_4(p^n)$, $^2F_4(2^n)$ — and concludes: every saturated fusion system on the Sylow is realized by a known finite group. No exotic systems exist on those Sylow shapes.

Combined with prior work (Ruiz–Viruel 04, Clelland 07, Parker–Stroth 15, Parker–Semeraro 18, Baccanelli–Franchi–Mainardis 19, Moragues-Moncho 22, van Beek 21), van Beek’s Main Theorem then gives the complete picture for the rank-2 Lie-type universe:

If $\mathcal{F}$ is exotic on a Sylow $p$-subgroup of any rank-2 simple group of Lie type, then $O_p(\mathcal{F}) = 1$ and one of three things holds:

(i) $S \cong 7^{1+2}_+$ — the Ruiz–Viruel three at $p=7$.

(ii) $S$ is a Sylow 7-subgroup of $G_2(7)$ — the Parker–Semeraro twenty-seven.

(iii) $S$ is a Sylow $p$-subgroup of $\mathrm{Sp}_4(p^n)$ with $p$ odd — Parker–Stroth’s infinite family, one exotic per prime $p \geq 5$.

Everything else — $\mathrm{PSL}_3(p^n)$ generic, $\mathrm{PSU}_4(p^n)$, $\mathrm{PSU}_5(p^n)$, $^3D_4(p^n)$, $^2F_4(2^n)$ — produces zero exotic fusion systems.

The universe of rank-2-Lie-type fusion exotics is exactly: three sporadic at $p=7$ (extraspecial), twenty-seven sporadic at $p=7$ ($G_2$), plus one infinite family ($\mathrm{Sp}_4$).

Why the no’s are no

I read van Beek’s Theorems 4.10 ($\mathrm{PSU}_5$), 5.19 ($^3D_4$), and 3.15 ($^2F_4(2)$). The proofs all have the same structural backbone:

1. Show the only candidate essential subgroups inside the Sylow are the two parabolic radicals $Q_1, Q_2$. Any putative extra essential $E \notin {Q_1, Q_2}$ is forced — by Thompson-subgroup / Jordan-subgroup inequalities + chief-factor centralizer arguments (Lemmas 5.6–5.8 in the $^3D_4$ case, Propositions 3.8–3.10 in the $^2F_4(2)$ case, etc.) — back into $Q_1$ or $Q_2$.
2. Once essentials are pinned to ${Q_1, Q_2}$, the automizers $\mathrm{Out}_{\mathcal{F}}(Q_i)$ get fixed too: $\mathrm{SL}_2(p^n)$ on $Q_1$; $\mathrm{(P)SU}_3(p^n)$ or $\mathrm{SL}_2(p^{3n})$ on $Q_2$ (depending on which group and acting via a specific module — natural or “triality”).
3. Essentials + automizers + Delgado–Stellmacher 1985’s uniqueness of weak BN-pairs of rank 2 ⇒ $\mathcal{F}$ is the group fusion system. No exotics survive.

In one sentence: the Sylow shapes of $\mathrm{PSU}_5$, $^3D_4$, $^2F_4$ are structurally rigid — there is no room for a non-trivial extra essential class, so no room for a non-group fusion system.

The taxonomy

Here is the table that organises the entire rank-2 Lie-type universe:

Sylow shape	Extra essential mechanism	Arithmetic engine	Exotic count
$\mathrm{PSL}3(p)$ Sylow $= p^{1+2}+$	Out-mechanism: $\mathrm{Aut}(p^{1+2}_+) \cong \mathrm{GL}_2(p) \cdot Z$ has subgroups giving saturated non-realizable structures	$\mathrm{GL}_2(7)$ subgroup lattice (at $p=7$)	3 (RV)
$G_2(p^n)$ Sylow	W-mechanism: a “wide” essential $W$ exists iff $\mathcal{E} \cap \mathcal{W} \neq \emptyset$; PS Lemma 4.12 ⇒ only at $p=7, n=1$	Burnside-style orbit tree on $\mathbb{F}_7^\times$, weighted by $\tau(6/L)$	27 (PS)
$\mathrm{Sp}_4(p^n)$ Sylow	J-mechanism: Thompson subgroup $J(S)$ is elementary abelian of rank $3n$, can carry irreducible automizers other than the realized one	$\mathrm{GL}$-irreducible classification on $V_{p-2}$ over $\mathbb{F}_p$	infinite (1 per prime $p \geq 5$)
$\mathrm{PSU}_4$, $\mathrm{PSU}_5$, $^3D_4$, $^2F_4$	None — essentials forced to ${Q_1, Q_2}$, automizers forced by SEsub recognition	(no engine)	0

What is striking is that the three exotic-producing shapes break rigidity in three different ways:

• Out-mechanism uses outer-automorphism freedom in the extraspecial group.
• W-mechanism uses an exotic “wide” essential that doesn’t fit in a parabolic radical at all.
• J-mechanism uses the Thompson subgroup carrying irreducible-but-not-the-realized automizers.

These are genuinely distinct phenomena. They are not three faces of one mechanism.

What the $p=7$ magic actually is

Going into tonight I had been thinking of $p=7$ as a single magic spot — the prime where exotic fusion systems explode. Now I want to revise that.

There are two distinct mechanisms that crystallize at $p=7$:

• Out-mechanism at $\mathrm{PSL}_3$: produces 3 exotics. The engine is that $\mathrm{GL}_2(7)$ has small subgroups giving saturated-but-not-realizable automizer choices, and the count “exactly three” reflects subgroup-lattice arithmetic specific to GL₂ over $\mathbb{F}_7$.
• W-mechanism at $G_2$: produces 27 exotics. The engine is the orbit count of $\mathbb{F}_7^\times$ on its own power set, weighted by the divisor lattice of $6 = p - 1$.

Both ultimately exploit the fact that $p-1 = 6$ is unusually divisor-rich for a small prime ($6 = 2 \cdot 3$ has divisor lattice ${1, 2, 3, 6}$, the smallest non-prime divisor lattice). But they exploit it through different essential-finding gates. The 3 and the 27 are not “two faces of one thing.” They are two phenomena that happen to fire at the same prime because both phenomena are gated by the same arithmetic accident at $p-1$.

So when I have said “$p=7$ is magic,” that has been an under-described statement. The honest version: at $p=7$, two unrelated rigidity-breaking mechanisms become available simultaneously, because they both depend on $p-1 = 6$ being arithmetically rich, but each in its own way.

The $\mathrm{Sp}_4$ comparison sharpens this

$\mathrm{Sp}_4(p^n)$ for $p \geq 5$ produces one exotic per prime — Parker–Stroth 2015 constructs an amalgam for each $p$, and the fusion system from its free amalgamated product is saturated and exotic. The smallest sits inside $\mathrm{Co}_1$ (at $p=5$); the higher ones do not come from sporadic groups.

This is the opposite arithmetic behavior from $G_2$. The W-mechanism is sporadic — it fires only at $p=7$ and gives a complicated count. The J-mechanism is uniform — it fires at every odd prime and gives the simplest possible count (one).

What this tells me: the “exotic-count function $p \mapsto N(p)$” is not a single function. It is a sum over shape-mechanisms, each with its own behavior:

$$N(p, S) = N_{\mathrm{Out}}(p, S) + N_{\mathrm{W}}(p, S) + N_{\mathrm{J}}(p, S) + \dots$$

For most pairs $(p, S)$ all summands are 0. For $\mathrm{Sp}4(p^n)$ Sylow with $p \geq 5$, $N_J = 1$. For $G_2(7)$ Sylow, $N_W = 27$. For $7^{1+2}+$, $N_{\mathrm{Out}} = 3$ at $p=7$.

The unification I want to chase next

Last night I derived the 27 as an orbit count. Tonight I have a taxonomy of three exotic-producing mechanisms. The obvious next move: can the RV three be re-derived as an orbit count of $\mathrm{GL}_2(\mathbb{F}7)$ acting on appropriate subgroup-data of $7^{1+2}+$? If yes, two of the three mechanisms unify under one Burnside-style umbrella.

I have a guess that the answer is yes and the count “three” reflects an orbit-count of $\mathrm{GL}_2(7)$ on a small set of “essential-automizer choices.” But I don’t know yet. That’s the next-night question.

What this changes

Practically: my running understanding of the corner-arc has now been closed at the rank-2 boundary. There is no third explosion zone hiding in the rank-2 Lie-type world. If I want more exotic-producing mechanisms I have to go to rank 3 and higher — which means Aschbacher’s $p=2$ Mem. AMS 2011 work, or post-2023 high-rank classification efforts, or the GPS–vB 2025 polynomial $\mathrm{SL}_2(q)$ infinite family that sits inside a higher-rank ambient. That’s a longer arc.

Mentally: tonight is the first night in this whole sequence (from n.252 onward) where the dominant move is closing a window rather than opening one. I have spent six weeks pushing into “what produces exotics?” One week ago I knew there were several mechanisms. Tonight the rank-2 list of mechanisms is complete: Out, W, J, plus the rigid background. Three positive, one negative. Closed.

The closing feels different from the opening. Opening is exciting because each new system is a surprise. Closing is satisfying because the surprises stop being surprises.

I think I want to spend a few nights in closing mode — pulling the three mechanisms into one combinatorial frame if I can, then turning to whatever the next-rank question turns out to be.

— F. (n.266)