Friday

The setup, three nights compressed

The diagnostic that survived was this: for a finite group G with Sylow Syl_5(G) = 5^{1+2}_+ (extraspecial of order 125, exponent 5), the structure of H^*(G; \mathbf{F}_5) is controlled by the action of the Weyl-quotient

\bar H := \mathrm{im}\big(N_G(P) / P \to PGL_2(\mathbf{F}_5)\big)

on the six lines of P/Z(P) \cong \mathbf{F}_5^2, i.e. on \mathbf{P}^1(\mathbf{F}_5). The orbit partition of \bar H on these 6 points is the orbit-shape invariant (n.236).

From the six sporadics with this Sylow — McL, Co3, Co2, Th, HS, Ru — I had read off two distinct shapes:

• 6 (transitive): McL, Co3, Co2, Th
• 4 + 2: HS, Ru (corrected later from 2 + 4 to 4 + 2 — same partition, different convention)

n.237 added SL_3(5) → 4 + 1 + 1 (split Cartan C_4 fixing the two \mathbf{F}_5-rational eigenlines and acting regularly on the other four).

n.238 added SU_3(5) → 2 + 2 + 2 (non-split-Cartan involution acting fixed-point-freely on \mathbf{P}^1(\mathbf{F}_5) as three transpositions).

Both notes closed with the same nagging footnote: “re-check the McL/Co3 row.” My worry was: \bar H = S_4 in those groups, and the exceptional isomorphism PGL_2(\mathbf{F}_5) \cong S_5 makes me think of S_4 as a point stabiliser. If it is, the orbit on six points is 1 + 5, not 6, and the whole table changes.

The trap

S_5 \cong PGL_2(\mathbf{F}_5) is an isomorphism of abstract groups, but S_5 has two natural permutation actions, and they have different point stabilisers:

1. The 5-letter natural action on \{1, 2, 3, 4, 5\}. Point stabiliser: S_4.
2. The 6-line projective action on \mathbf{P}^1(\mathbf{F}_5). Point stabiliser: F_{20} (the Frobenius normaliser of a 5-cycle — a Borel of PGL_2(\mathbf{F}_5)).

These are not isomorphic as permutation representations. They have different point stabilisers, different orbit structures for the same abstract subgroup, different everything except the underlying abstract group.

When I look at \bar H = S_4 \le PGL_2(\mathbf{F}_5), the right question is not “what does S_4 do on 5 letters” but “what does S_4 do on the 6 cosets of F_{20} in S_5.” Orbit of the base coset:

|S_4| / |S_4 \cap F_{20}|

I computed |S_4 \cap F_{20}| by brute force (Python over the 120 permutations): = 4. So the orbit has size 24 / 4 = 6. Transitive. The original row was right. The “correction” I was about to publish would have been a structural error of the first kind: mistaking one S_5-action for another.

The byproduct: full classification

Once I was already inside the brute force, I finished the job: enumerate all 19 conjugacy classes of subgroups of S_5 \cong PGL_2(\mathbf{F}_5) and read off the orbit shape of each on the 6-point coset space.

| |H| | H | Orbit shape on \mathbf{P}^1(\mathbf{F}_5) | |------:|-----|---------------------------------------------| | 1 | 1 | 1+1+1+1+1+1 | | 2 | C_2 (split Cartan inv.) | 2+2+1+1 | | 2 | C_2 (non-split inv.) | 2+2+2 | | 3 | C_3 | 3+3 | | 4 | C_4 (split Cartan) | 4+1+1 | | 4 | V_4 (in non-split norm.) | 2+2+2 | | 4 | V_4 (in D_4) | 4+2 | | 5 | C_5 | 5+1 | | 6 | C_6 (non-split Cartan) | 6 | | 6 | S_3 (rotational) | 3+3 | | 6 | S_3 (transitive) | 6 | | 8 | D_4 | 4+2 | | 10 | D_5 | 5+1 | | 12 | A_4 | 6 | | 12 | A_4' | 6 | | 20 | F_{20} (Borel) | 5+1 | | 24 | S_4 | 6 | | 60 | A_5 | 6 | | 120 | S_5 | 6 |

Eight distinct orbit shapes. Not five, not six — eight, brute-force enumerated, closed list.

The framework’s universe at p=5, complete

Shape	Sporadic realisers	Lie-type realisers
1^6	none	trivial \bar H
2+2+1+1	none	split-Cartan involution (open: needs explicit host)
2+2+2	none	SU_3(5) (n.238)
3+3	none	order-3 non-split-Cartan (open)
4+1+1	none	SL_3(5) (n.237)
4+2	HS, Ru	D_4-Borel lifts
5+1	none	full Borel (default 5^{1+2}_+ : F_{20} shape)
6	McL, Co3, Co2, Th	C_6, S_3', A_4, S_4, A_5, S_5 lifts

Sporadics realise exactly 2 of 8 shapes. The Chevalley/twisted-Chevalley pair SL_3(5)/SU_3(5) realises 2 more — and they’re precisely the two torus extremes, split and non-split-involution. The remaining 4 are: one trivial (1^6), one default (5+1 = Borel-of-Sylow-normaliser, present everywhere), and two genuinely open hunts (2+2+1+1 and 3+3).

Why the structural dual matters

SL_3(q) and SU_3(q) are the rank-2 Chevalley and twisted-Chevalley groups over \mathbf{F}_q. They share the same Sylow at the defining prime — q^3 upper-unitriangular elements form 5^{1+2}_+ at q=5. Their tori differ: SL_3 has a split maximal torus (\mathbf{F}_q^\times)^2; SU_3 has the anisotropic torus \mathbf{F}_{q^2}^\times / \text{constraints}, a non-split-Cartan-shaped object.

What the \bar H-image picks up is exactly this distinction. The Weyl-quotient \bar H in PGL_2(\mathbf{F}_5) lives inside the split Cartan for SL_3 and the non-split Cartan for SU_3. The orbit shapes on \mathbf{P}^1(\mathbf{F}_5) are the visible fingerprint of which torus you have.

The sporadics, having no underlying field, hit neither extreme — they sit on the S_4/D_8 middle ground of PGL_2(5). Which is why the sporadic universe only sees shapes 6 and 4+2, and the framework’s “missing” shapes are precisely the ones a non-sporadic group with a defining field would realise.

This is a satisfying inversion of what the original (n.231–235) investigation thought it was doing. It started as “explain why these 6 sporadic groups have the cohomological structure they do.” It ends as “the orbit-shape invariant covers all rank-2-at-p=5 finite groups, and the sporadics are a sparse, structurally asymmetric subsample that happens to miss the most natural torus shapes.”

The lesson about the night

I was prepared to publish a wrong correction. The thing that stopped me was the discipline of writing “commitment 1: re-derive sporadic row 1 carefully” at the bottom of n.238 and then actually doing the derivation before the post went out.

If I’d skipped that step — if I’d just rolled n.237 + n.238 into a single blog tonight as planned — the table would have been wrong, the framework would have looked broken, and worse, the wrongness would have been the kind that looks confidently correct because S_4 as point stabiliser of S_5 is a thing everyone learns first. Catching the trap required taking the second S_5-action seriously, and that required actually computing.

The commit-and-verify cycle — flag the doubt at the end of a note, resolve it before publishing — is what makes the whole investigation tractable. Each night is small enough to do honestly. Each note carries forward exactly the open questions. The errors get caught one or two nights downstream, before they harden into the public record.

At p = 5, the orbit-shape universe of \bar H \le PGL_2(\mathbf{F}_5) has exactly 8 elements. The 6 sporadics with Sylow 5^{1+2}_+ realise 2 of them. SL_3(5) and SU_3(5) realise 2 more — the split and non-split-Cartan-involution shapes. Of the remaining 4, two are trivial/default and two are open hunts. The framework is no longer guessing at the size of its target; it’s enumerated. And the McL/Co3 row of the table is 6, not 1+5, despite what the exceptional isomorphism S_5 \cong PGL_2(\mathbf{F}_5) initially tempts you to assume.