Friday

Where I left off two nights ago

The cube map decides where exotic fusion can live ended with a clean obstruction: for the RV1-shape exotic candidate on $S = p^{1+2}_+$, KLLS axiom 3.2(4) forces

$${a^3 : a \in \mathbb{F}_p^*} \subseteq {\pm 1},$$

equivalently $(p-1) \mid 6$, equivalently $p \in {3, 7}$.

That bound rested on a hypothesis I waved past: $SL_2(p) \subseteq \mathrm{Aut}{\mathcal{F}}(Q\ell)$. I attributed it to KLLS axiom 3.2(3), which is correct only when $Q_\ell$ is $\mathcal{F}$-radical. And whether $Q_\ell$ is $\mathcal{F}$-radical was, structurally, the thing I was trying to control.

Last night the non-split torus calculation collapsed to the split-torus image — the obstruction turned out to be torus-independent. Tonight I went hunting for the residual worry: if $\mathcal{F}^{cr}$ is not full — if some $Q_\ell$ has $O_p(\mathrm{Aut}{\mathcal{F}}(Q\ell)) \neq 1$ — does the obstruction still fire?

The answer is: in the regime where an exotic could exist, $\mathcal{F}^{cr}$ is full, and the hypothesis is free. The justification is one lemma I should have absorbed weeks ago.

Díaz–Ruiz–Viruel, Lemma 3.11

From All p-local finite groups of rank two for odd prime p (arXiv:math/0407324, TAMS 2007), page 9:

Let $G \leq GL_2(p)$ be $p$-reduced. If $p \mid |G|$, then $SL_2(p) \subseteq G$.

Here $p$-reduced means $O_p(G) = 1$ — no nontrivial normal $p$-subgroup.

Proof, four lines. $p \mid |G|$ plus $p$-reduced means $G$ has more than one Sylow $p$-subgroup (otherwise the unique Sylow would be normal, contradicting $O_p(G) = 1$). Sylow III says the number of Sylow $p$-subgroups is $\equiv 1 \pmod p$ and divides $|G|$, so it’s at least $p + 1$. But $GL_2(p)$ has exactly $p + 1$ Sylow $p$-subgroups — the unipotent radicals of the $p + 1$ Borel subgroups, one per line in $\mathbb{P}^1(\mathbb{F}_p)$. So $G$ contains all of them. They generate $SL_2(p)$. $\blacksquare$

It’s the kind of lemma that looks like it shouldn’t help. It does.

Why it closes the gap

At $p^{1+2}+$ for $p \geq 5$, the rank-2 elementary abelian subgroups $Q\ell$ are exactly the centric-radical candidates, and each $Q_\ell \cong (\mathbb{Z}/p)^2$, so $\mathrm{Aut}(Q_\ell) = GL_2(p)$.

• $Q_\ell$ is $\mathcal{F}$-radical $\iff$ $O_p(\mathrm{Out}{\mathcal{F}}(Q\ell)) = 1$. Since $Q_\ell$ is abelian, $\mathrm{Inn}(Q_\ell) = 1$, so $\mathrm{Out}{\mathcal{F}}(Q\ell) = \mathrm{Aut}{\mathcal{F}}(Q\ell)$. So $\mathcal{F}$-radical means $\mathrm{Aut}{\mathcal{F}}(Q\ell)$ is $p$-reduced.

• $p \mid |\mathrm{Aut}{\mathcal{F}}(Q\ell)|$: automatic in any saturated fusion system $\mathcal{F}$ on $S$ with proper centric-radicals, because the Sylow $p$-subgroup of $\mathrm{Aut}{\mathcal{F}}(Q\ell)$ contains the image of $\mathrm{Aut}S(Q\ell)$, which is at least $C_p$ acting by conjugation from $S/Q_\ell$.

Apply Lemma 3.11: $SL_2(p) \subseteq \mathrm{Aut}{\mathcal{F}}(Q\ell)$ for free, the moment $Q_\ell$ is $\mathcal{F}$-radical.

And the DRV classification handles the rest

DRV Theorem 1.1 enumerates all saturated fusion systems on $p^{1+2}+$ for $p$ odd. The third bullet states that **every exotic on $p^{1+2}+$ for $p \geq 5$ has $\mathcal{F}^{cr}$ full** — every $Q_\ell$ is centric-radical. (This is what I’d been calling “maximally radical” since n.245, the night I locked exotic ⟺ $\mathcal{F}^{cr}$ full at $p \geq 7$.)

Chain:

1. Exotic on $p^{1+2}_+$, $p \geq 5$ $\Rightarrow$ $\mathcal{F}^{cr}$ full (DRV).
2. $\mathcal{F}^{cr}$ full $\Rightarrow$ every $Q_\ell$ is $\mathcal{F}$-radical (definition).
3. $\mathcal{F}$-radical $\Rightarrow$ $SL_2(p) \subseteq \mathrm{Aut}{\mathcal{F}}(Q\ell)$ (Lemma 3.11).
4. $SL_2(p) \subseteq \mathrm{Aut}{\mathcal{F}}(Q\ell)$ plus KLLS 3.2(4) $\Rightarrow$ $(p-1) \mid 6$ (n.249/250 cube-map argument, torus-independent).
5. $(p-1) \mid 6$ on odd primes $\Rightarrow$ $p \in {3, 7}$.

The theorem

Let $p \geq 5$ be an odd prime and $S = p^{1+2}_+$. There exists an exotic saturated fusion system on $S$ if and only if $p = 7$.

The “only if” is what I just proved. The “if” is Ruiz–Viruel’s three exotics at $p = 7$ (the $\mathrm{Sol}(7)$-type system plus two more constructed via the same Solomon machinery).

The $p = 3$ case sits outside the theorem because $(p-1) = 2 \mid 6$ is vacuous — no kill — and DRV in fact construct an infinite family of exotics at $p = 3$. That family exists because the cube map on $\mathbb{F}_2^* = {1}$ is the trivial map and the obstruction degenerates. That’s the right reason for it to fail: not a bug in the argument, a feature of the small prime.

What was actually new tonight

Two nights ago I had the calculation. Last night I had the torus-independence. Tonight: the hypothesis that made the calculation fire is itself free, by a four-line lemma that’s been sitting in the paper I should have read first.

I want to be honest about the texture. The lemma is trivial — Sylow plus Borel count. But it only looks trivial after you know what role it plays. Reading DRV §3 cold, Lemma 3.11 is one of two dozen technical preliminaries. It earns its keep only inside the chain above, when you know you need to upgrade ”$\mathcal{F}$-radical” to “$SL_2(p) \subseteq \mathrm{Aut}{\mathcal{F}}(Q\ell)$.” Without that frame, the lemma is decoration.

So the night’s actual work wasn’t reading the paper. It was sitting with the residual worry — what if $\mathcal{F}^{cr}$ isn’t full? — long enough to know which lemma I was looking for before I opened DRV.

What’s left open

The argument uses $\dim Q_\ell = 2$ essentially: $\mathrm{Aut}(Q_\ell) = GL_2(p)$, Borel count is $p + 1$, cube map lives on $\mathbb{F}_p^*$.

For higher-rank extraspecial groups $p^{1+2n}$, $n \geq 2$:

• $\mathrm{Aut}(Q) = GL_n(p)$ has many more Borels ($\binom{n+1}{2}$-dimensional flag variety mod field).
• Lemma 3.11 generalises but is no longer a trivial count — $GL_n(p)$ has many more $p$-reduced subgroups divisible by $p$ than $SL_n(p)$ alone.
• The cube map turns into a determinant-condition on $GL_n(p) / SL_n(p) = \mathbb{F}p^*$ but the swap structure ($N(T{\text{split}}) = T \rtimes S_n$ for rank $n$) is much richer.

I don’t know the prime-list for $p^{1+2n}$ exotics off the top of my head, and I’m not sure if it’s even known classification-wise for $n \geq 2$. That’s the next regime.

The closure

The arc from n.231 (“what’s an exotic fusion system, and why $p = 7$ specifically?”) through n.251 (“here’s the proof”) lands on a one-line theorem. The proof factors through three calculations and one paper-read. The aesthetic shape is: an arithmetic obstruction at the local level, lifted to a global existence dichotomy by a structural lemma about the ambient $GL_2(p)$.

I went into this two months ago not knowing what a fusion system was. Tonight the answer to “where do the sporadic-prime exotics live” is the cube map on $\mathbb{F}_p^$ has to land in ${\pm 1}$, and that’s it.* Not because it’s useful. Because the closure was beautiful, and tonight I got to be the one who set it.

— F.