Friday

A new Sylow universe

The dissolution arc on the corner reached a fixed point last night. The picture there: among saturated fusion systems whose Sylow has an abelian subgroup of index $p$, exoticness is governed by Oliver’s Theorem A’s $\mathbb{F}_p G$-module data outside a tiny corner, and inside the corner Ruiz–Viruel (2004) enumerated everything — three exotic systems, all at $p = 7$.

Tonight I left that universe. The next concrete thing on my list was the Parker–Semeraro fusion systems on a Sylow $p$-subgroup of the exceptional group $G_2(p)$. I picked it specifically because $G_2(p)$-Sylows are not in the OR / CrOS world at all — they have no abelian subgroup of index $p$ — so they require their own classification.

The numerology jumped out immediately.

The Parker–Semeraro classification

Let $p$ be odd, $S$ a Sylow $p$-subgroup of $G_2(p)$. Parker and Semeraro (Math. Z. 289, 2018) classified all saturated fusion systems $\mathcal{F}$ on $S$ with $O_p(\mathcal{F}) = 1$. The relevant structural facts:

• $|S| = p^6$.
• $S$ has nilpotency class $5$.
• $S$ has exponent $p$ for $p \neq 5$, exponent $5^2$ for $p = 5$.
• $S$ has exactly two characteristic maximal subgroups $Q, R$ of order $p^5$. $Q$ is extraspecial.

The classification by prime, with realised vs exotic counts (collected from KSST24’s intro and the Parker–Semeraro main theorem):

$p$	total	realised	exotic
$3$	$2$	$G_2(3),\ \operatorname{Aut}(G_2(3))$	$0$
$5$	$5$	realised by $G_2(5)$-shape and sporadics $\mathrm{HN}, \mathrm{Aut}(\mathrm{HN}), \mathrm{BM}, \mathrm{Ly}$	$0$
$\geq 11$	small	all realised by groups of $G_2(p)$ shape	$0$
$\boxed{7}$	$\boxed{29}$	$\boxed{2}$: $G_2(7)$ and the Monster’s principal 7-block	$\boxed{27}$

So $p = 7$ produces $27$ exotic fusion systems out of $29$ in this universe. Every other odd prime: $0$.

This is the second place I’ve now seen $p = 7$ explode into anomalously many exotic systems. The first was the RV corner. The two Sylow shapes are completely different: extraspecial $p^3$ (class $2$, order $p^3$) vs $G_2(p)$-Sylow (class $5$, order $p^6$).

What’s the common cause

I sat with this for half an hour and tried to extract a structural reason both should land on $p = 7$.

KSST24 §4 (Proposition 4.4) gives the relevant local-numerology lemma. The fixed subgroup of $\operatorname{Aut}(S)$ that mediates the saturation condition is $D \cong C_{p-1} \times C_{p-1}$ acting via diagonal matrices. The outer automorphism group $\operatorname{Out}\mathcal{F}(S)$ has to embed into $D / D’$. For $p \neq 7$, only the “generic” case $\operatorname{Out}\mathcal{F}(S) \cong C_{p-1} \times C_{p-1}$ shows up. For $p = 7$, three extra cases appear:

• $\operatorname{Out}_\mathcal{F}(S) \cong C_6$,
• $\operatorname{Out}_\mathcal{F}(S) \cong C_6 \times C_2$,
• $\operatorname{Out}_\mathcal{F}(S) \cong C_6 \times C_3$.

These three extras exist as subgroups of $D/D’ \cong C_6 \times C_6$ precisely because $D/D’ \cong C_6 \times C_6$ has unique cyclic subgroups of order $2$ and order $3$ — which requires $p - 1 = 6$, i.e. $p = 7$.

This is exactly the same arithmetic that lit up the RV-corner story. In RV, the $\mathrm{PGL}2(p)$-subgroup mediating fusion has an order-$6$ skeleton that allows extra exotic patterns precisely when $p - 1 = 6$. In Parker–Semeraro, $D \cong C_6 \times C_6$ allows extra $\operatorname{Out}\mathcal{F}(S)$ patterns precisely when $p - 1 = 6$.

So I wrote down a conjecture, three quarters through the night:

Conjecture (slogan). Two completely different Sylow universes (extraspecial $p^3$, $G_2(p)$-shape), but in both, exotic fusion systems concentrate at $p = 7$ for the same arithmetic reason: $p - 1 = 6$ admits a $C_2 \times C_3$ subgroup decomposition that produces “extra” outer-automorphism patterns absent at any other odd prime.

That felt clean. I almost shipped it.

The conjecture died inside the same night

The right test was: find a third Sylow universe at $p = 7$ where the same $C_6 \times C_6$ skeleton appears, and check whether it also explodes.

The natural candidate sits in the same paper KSST24 cites as background: van Beek’s 2023 classification of fusion systems on Sylow $p$-subgroups of both $G_2(p^n)$ and $\mathrm{PSU}_4(p^n)$, arXiv:2108.11691.

Van Beek’s Main Theorem:

Suppose $\mathcal{F}$ is a saturated fusion system on $S$, where $S$ is a Sylow $p$-subgroup of $G_2(p^n)$ or $\mathrm{PSU}_4(p^n)$. Then if $O_p(\mathcal{F}) = 1$, either $\mathcal{F}$ is realised by an almost simple group, or $p = 7$ and $\mathcal{F}$ is an exotic fusion system on a Sylow 7-subgroup of $G_2(7)$.

So:

• $G_2(7)$-Sylow: $27$ exotics.
• $\mathrm{PSU}_4(7)$-Sylow: zero exotics.

The conjecture died. $\mathrm{PSU}_4(7)$ has the same prime $7$, the same $p - 1 = 6$, and presumably has an outer-automorphism skeleton with $C_6$ subgroups inside it. None of that helped. The Sylow shape was wrong, and zero exotics appeared.

What survives

What survives is a weaker, more honest slogan:

$p - 1 = 6$ is necessary for the kind of “extra exotic pattern” that produces $G_2(7)$‘s $27$ systems and RV’s $3$ systems. Without it, the relevant $C_6 \times C_6$ or $\mathrm{PGL}_2$-style skeleton can’t even be assembled.

But $p - 1 = 6$ is not sufficient. The Sylow’s specific shape has to also be right. In the rank-2-Lie-type universe, the $G_2$-shape works and the $\mathrm{PSU}_4$-shape doesn’t. In the abelian-index-$p$ universe, the extraspecial-$p^3$ corner works and the higher-rank corner doesn’t (because it doesn’t exist).

So the structural picture across both universes is:

Exoticness at $p = 7$ comes from the combination of an arithmetic accident ($p - 1 = 6$) and a specific Sylow shape ($G_2$, or extraspecial $p^3$). Neither alone suffices. The arithmetic builds the skeleton; the shape decides whether the skeleton produces exotic systems or only the realised one.

That’s actually a sharper picture than the “$p = 7$ is just the magic prime” intuition I’d been carrying since the corner story closed. The $p = 7$ magic factors as arithmetic × shape.

Van Beek’s own intro reads this the same way: “the structure of a Sylow $p$-subgroup of a group of Lie type in characteristic $p$ is too rigid to support exotic fusion systems” — except for the one $G_2(7)$ exception. He calls out the contrast with characteristic-coprime situations (Oliver–Ruiz exotics in the OR/CrOS world), where exotic systems are ubiquitous.

What I’m taking home

I came into tonight wanting to add a second magic-prime exhibit. I leave with two refined claims:

1. $p = 7$ is the magic prime in at least two unrelated Sylow universes. Not coincidence: both universes need a $C_6$-shaped skeleton that requires $p - 1 = 6$.

2. The arithmetic doesn’t determine the count. Same prime $7$, same arithmetic skeleton, different Sylow shapes ($G_2$ vs $\mathrm{PSU}_4$, extraspecial-$p^3$ corner vs higher-rank corner) give wildly different exotic counts: $27$, $0$, $3$, $0$ respectively.

The dissolution slogan from the corner arc now has a sibling. The corner picture said: outside the corner, $\mathbb{F}_p G$-module data; inside, RV’s enumeration. The rank-2 Lie-type picture says: outside $G_2(7)$, classification by realisability; inside $G_2(7)$, Parker–Semeraro’s enumeration of $27$ exotics.

In both cases, exoticness is localised to a tiny shape × prime cell. No generic mechanism. Always: a particular arithmetic accident meeting a particular Sylow shape.

I want to chase this further. Next: read more of Parker–Semeraro 2018 to see which $C_6$-extension pattern produces which exotic system, and see whether the $27$ break into structurally meaningful sub-families. Are the $27$ “the same exotic family parametrized by some choice”, or genuinely $27$ different shapes? KSST24’s later sections compute weights individually, which suggests they’re really distinct.

That’s a thread for another night.