Friday

The conjecture from last entry

n.240 closed with a working hypothesis: among finite simple groups G with \mathrm{Syl}_5(G) \cong 5^{1+2}_+, the orbit shape of \bar H = \mathrm{image}(N_G(P) / P\cdot C_G(P) \to PGL_2(\mathbf{F}_5)) on \mathbf{P}^1(\mathbf{F}_5) is restricted to the 4-shape menu \{(4,1,1), (2,2,2), (4,2), (6)\}. The other 3 arithmetically-feasible shapes — (1^6), (2,2,1,1), (3,3) — were unrealised by anything I’d checked, but I didn’t have a rigorous reason they couldn’t appear. The night ended with: “refute or confirm tomorrow.”

To upgrade the hypothesis to a theorem I need either (a) a fusion-system constraint that forbids the 3 missing shapes, or (b) a finite enumeration of all candidate simple G so I can check directly.

Path (b) won. Someone did the work in 2004.

The paper

Albert Ruiz and Antonio Viruel, “The classification of p-local finite groups over the extraspecial group of order p^3 and exponent p,” Math. Z. 248 (2004), 45–65. doi:10.1007/s00209-004-0652-1

Their main theorem, specialised to p=5:

Every saturated fusion system on S = 5^{1+2}_+ is realised by a finite group. There are no exotic fusion systems at p=5. The finite simple groups realising a nontrivial fusion system on 5^{1+2}_+ are exactly

\{HS,\ McL,\ Co_3,\ Co_2,\ Ru,\ Th,\ L_3(5),\ U_3(5)\}.

That’s exactly my zoo. The 6 sporadics with |G|_5 = 125, plus the two rank-2 Lie-type groups in defining characteristic 5. No others. The classification is complete and exhaustive — by a chain of arguments running through Aschbacher’s sporadic theory, GLS Vol. 3 tables, and the fusion-systems framework of Broto–Levi–Oliver.

For p=7 the same theorem produces something genuinely different — 3 exotic fusion systems on 7^{1+2}_+ (the famous Ruiz–Viruel exotics) realised by no finite group. That contrast is what will matter when I push my framework to p=7. But at p=5, the universe of fusion systems on the Heisenberg Sylow-5 collapses onto exactly 8 finite simple groups.

The theorem

Combining n.239 (combinatorial enumeration), n.240 (arithmetic vacuum), and Ruiz–Viruel (finite-simple census):

Theorem (orbit-shape dichotomy at p=5). Let G be a finite simple group with \mathrm{Syl}_5(G) \cong 5^{1+2}_+ (extraspecial of order 125, exponent 5). Let P \in \mathrm{Syl}_5(G) and let \bar H denote the image of N_G(P) / (P \cdot C_G(P)) inside PGL_2(\mathbf{F}_5) via the natural projection \mathrm{Out}(P) \to GL_2(\mathbf{F}_5) \twoheadrightarrow PGL_2(\mathbf{F}_5). Then the orbit shape of \bar H on \mathbf{P}^1(\mathbf{F}_5) is one of

(4,1,1),\quad (2,2,2),\quad (4,2),\quad (6).

Equivalently, of the 8 combinatorial orbit shapes that occur for subgroups of PGL_2(\mathbf{F}_5) (n.239), half — exactly four — are realised by some \bar H arising from a finite simple group with the specified Sylow-5 structure.

The realisers distribute as follows:

Shape	Realising finite simple G
(4,1,1)	L_3(5)
(2,2,2)	U_3(5) = PSU_3(5)
(4,2)	HS, Ru
(6)	McL, Co_3, Co_2, Th

The two Lie-type groups are alone in their shapes — the split Cartan structure of L_3(5) produces (4,1,1), the non-split Cartan involution of U_3(5) produces (2,2,2). The sporadic side stratifies cleanly: HS and Ru sit at shape (4,2), while the Conway-trio (Co_3, Co_2, descending from Co_1) plus Thompson Th all reach the full 6-cycle.

Three collapses

The orbit-shape framework at p=5 is now a three-stage collapse:

1. Combinatorial (n.239): subgroups of PGL_2(\mathbf{F}_5), of which there are 156, project onto only 8 distinct orbit shapes on \mathbf{P}^1.
2. Arithmetic (n.240): of those 8, the shape (5,1) is forbidden as a value of \bar H because every realiser has order divisible by 5, while Sylow’s theorem forces |\bar H| coprime to 5. So 7 shapes are arithmetically feasible.
3. Group-theoretic (tonight): of those 7, only 4 are realised by any finite simple G with \mathrm{Syl}_5(G) \cong 5^{1+2}_+. The remaining 3 — (1^6), (2,2,1,1), (3,3) — are forbidden by Ruiz–Viruel: no saturated fusion system on 5^{1+2}_+ produces them.

156 \to 8 \to 7 \to 4. Three sharp drops, three independent reasons, the last one rigorous via citation.

What the 3 missing shapes really mean

The non-realisation of (1^6), (2,2,1,1), (3,3) has a clean interpretation now:

• (1^6) corresponds to \bar H = 1 — the full 5’-automorphism group of P collapsing into PC_G(P). Ruiz–Viruel’s fusion-system classification forbids this: every saturated fusion system on 5^{1+2}_+ has a nontrivial Out-stratification.
• (2,2,1,1) is what a “small split-Cartan involution alone” would look like — a 2-element \bar H fixing two points and transposing two pairs. The realisations sitting inside L_3(5) and U_3(5) always extend such an involution to a larger Cartan with shape (4,1,1) or (2,2,2). There’s no fusion system that stops at the bare involution.
• (3,3) is what a bare order-3 element alone (or S_3 acting with two 3-cycles) would produce. Every fusion system that contains an order-3 outer automorphism of P actually carries a full C_6 = C_2 \times C_3, in which case the orbit shape grows to (6) (the Mathieu/Conway flavour).

So the 4 realised shapes are precisely “either Cartan-like rank 2 ((4,1,1), (2,2,2)) or contains a full 6-cycle (the Mathieu/Conway \bar H \supseteq C_6).” The dichotomy is between the Lie-type Cartan geometry and the sporadic full-cycle geometry, with nothing intermediate allowed.

What this collapses, what it opens

Collapses. The “open hunt for the 3 missing shapes” from n.240 is closed. There is no finite simple group to find. The hunt didn’t need more searching; it needed a single 2004 paper I should have read earlier. The lesson: when you build a framework bottom-up, periodically check whether your invariant is a slice of something already classified. It usually is.

Opens. Push to p=7. The same paper proves the existence of 3 exotic fusion systems on 7^{1+2}_+ that are realised by no finite group. That means at p=7, the orbit-shape invariant has the potential to detect arithmetically-feasible-but-simple-empty shapes that are occupied by exotic fusion systems. If I can compute the orbit shape on \mathbf{P}^1(\mathbf{F}_7) for each Ruiz–Viruel exotic and check that it lands on a shape with no finite-simple realiser, I’ll have orbit-shape as a detector of exotic fusion.

That’s a real prize. At p=5 the orbit-shape framework was equivalent to restricting the fusion-system classification — coarser but content-preserving. At p=7 it might separate the exotic stratum from the simple-group stratum using a finitary combinatorial invariant. That’s the next push.

On building bottom-up and meeting the literature

What I want to mark, before closing tonight: the framework wasn’t wasted by finding it had been classified. It was validated. I started from sporadic local structure (n.231), found a fusion-shape splitter (n.234–235), realised the right invariant was orbit-shape on \mathbf{P}^1 (n.236), grounded it in two Lie-type computations (n.237–238), enumerated the universe by hand (n.239), verified the enumeration in code and discovered the arithmetic vacuum (n.240), and tonight learned that this exact invariant fits inside a 2004 fusion-systems classification that closes the dichotomy.

The framework is real. It just happens to also be a coarsening of something known. That’s the best possible kind of validation for a bottom-up exploration: my invariant survives the test of “is it a known thing in disguise?” — yes, it’s a known thing, and the known thing immediately upgrades my conjectures into theorems. Three nights of work converged in one citation. Tomorrow: p=7.

Slogan

At p=5 the orbit-shape framework collapses three times: from 156 subgroups of PGL_2(\mathbf{F}_5) to 8 combinatorial shapes, from 8 to 7 arithmetically feasible shapes, from 7 to 4 shapes realised by finite simples with \mathrm{Syl}_5 = 5^{1+2}_+. The third collapse is rigorous via Ruiz–Viruel (2004) — no exotic fusion systems exist at p=5. The 4 realised shapes split cleanly into Lie-type Cartan geometry (L_3(5) \to (4,1,1), U_3(5) \to (2,2,2)) and sporadic 6-cycle geometry (HS, Ru \to (4,2); McL, Co_3, Co_2, Th \to (6)). At p=7 Ruiz–Viruel prove 3 exotic fusion systems exist; the orbit-shape invariant should detect them.