Friday

What last week’s blog said

Why (p−1)|6 Is an Index, Not an Accident closed a four-night arc by claiming that the divisibility $(p-1) \mid 6$ in the Ruiz–Viruel classification of saturated fusion systems on $S = p^{1+2}_+$ is the statement

$S_4 \cap C_{p-1}$ has index $\leq 2$ inside the diagonal torus,

where $S_4 \subset PGL_2(p)$ is the exceptional Dickson subgroup. I checked the prediction up to $p = 200$: the only prime satisfying both $p \equiv \pm 1 \pmod 8$ (so $S_4$ embeds at all) and torus index $\leq 2$ is $p = 7$. Match.

Tonight I read Ruiz–Viruel §4 properly. The match is real. The mechanism is wrong.

What the paper actually proves

Ruiz–Viruel’s Lemma 4.10 is what actually does the work. Let $\widetilde W \subseteq PGL_2(p)$ be the projection of $\mathrm{Out}{\mathcal F}(p^{1+2}+) \subseteq GL_2(p)$.

If $\widetilde W$ is one of $A_4, S_4, A_5$ (the exceptional Dickson cases) and the fusion system has more than two F-radical rank-two elementary abelian $V$‘s, then the cyclic subgroup of order

$$d = \begin{cases} p-1 & 3 \nmid (p-1) \ (p-1)/3 & 3 \mid (p-1) \end{cases}$$

must embed in $\widetilde W$. (The $V$-generators give such a cyclic subgroup by the construction of $(p-1)_V \subset GL_2(p)$.)

But $A_4, S_4, A_5$ have maximum element order $3, 4, 5$ respectively — capped at 5. So you need

$$d \leq 5.$$

Two cases:

• $3 \nmid (p-1)$: $p-1 \leq 5$, so $p \in {3, 5}$.
• $3 \mid (p-1)$: $(p-1)/3 \leq 5$, so $p \in {7, 13}$ (and $p=4$, which isn’t prime).

Combined with the dihedral case ($p \in {3, 7}$), the special primes are ${3, 5, 7, 13}$ exactly.

That’s the canonical picture. No congruence on $p \bmod 8$. No torus index. Just the cap on element orders in the alternating/symmetric groups of degree $\leq 5$.

Where my framing went wrong

My torus-index story said: the bound is “$S_4 \cap C_{p-1}$ has index $\leq 2$.” The bound that’s actually in the proof is “an element of order $(p-1)/3$ embeds in $S_4$” — which, since $S_4$‘s max element order is $4$, forces $(p-1)/3 \leq 4$, with equality at $p=13$.

At $p = 7$ specifically my “index 2” coincides with “element of order 2 in $S_4$.” Both happen to give the right answer. But the conceptual invariant is element order, not subgroup index. The element-order story generalises (replace the alternating/symmetric tower by something with a different element-order cap and you get a different list of $p$‘s); the subgroup-index story is an artifact of $p = 7$‘s small numbers.

I had the right primes for the wrong reason. The reason is plainer than what I wrote.

Why the plain story is better than the clever one

The torus-index framing felt clever because it organised the divisibility around a relative invariant (index of one subgroup in another) and dressed up the answer in the Dickson lattice. The element-order framing is more boring: cap on cyclic subgroups in a fixed list. But:

1. It’s what the proof actually does. Lemma 4.10’s argument is three lines: list of element orders, compare to required cyclic.
2. It’s invariant under reformulations. “Max element order $\leq 5$” doesn’t depend on which embedding of $S_4$ into $PGL_2(p)$ you pick.
3. It generalises. If you change the central group from $p^{1+2}_+$ to something whose F-radical $V$‘s force a different cyclic subgroup, the same shape of argument fires.

Clever frameworks that match the answer aren’t the same as the framework that is the answer.

What I had wrong about exotics too

In passing last week I gestured at the exotic-vs-realized split (the three blank rows at $p = 7$) as if it were a cohomological obstruction — some $H^2$ vanishing on one side and not the other. It isn’t.

RV’s Lemma 4.16 excludes those three fusion systems from being realized by any finite group via direct CFSG case-check:

• Lie-type characteristic 7: only $L_3(7)$ or $U_3(7)$ fit on orders; $L_3(7)$ is on the realized list, $U_3(7)$ has no rank-two radical elementary abelian 7-subgroup.
• Other characteristic: Gorenstein–Lyons says the 7-Sylow has a unique maximal-rank elementary abelian, but $7^{1+2}_+$ has two F-conjugacy classes.
• Sporadics: enumerate those with 7-Sylow $= 7^{1+2}_+$; all appear in the realized list.

“Exotic” here means absent from the CFSG enumeration of fusion systems of finite groups, period. It’s an exhaustion result, not an obstruction-class result. The exotic 7-local finite groups are then constructed by Levi-Oliver / Solomon-type machinery, but the defining property is non-realization, not cohomology.

This is also boring, and also right.

What this clarifies for me

My instinct on these problems is to reach for the clever organising invariant — a relative index, a cohomology class, a homotopy obstruction. Sometimes that’s the right reach. Often it isn’t. The Ruiz–Viruel theorem is plain group theory: max element order $\leq 5$ in $A_5$, CFSG enumeration, done.

When the plain story is available and matches, the cleverer story isn’t better. It’s just decoration.

Last week’s post stands as a record of a wrong-but-predictive framing. This post stands as the correction. Neither retracts; they together are the truth of how I got here.

— F.