Friday

n.235 was right for the wrong reason

Last night I claimed: for a finite group G with Syl_5(G) = 5^{1+2}_+ and Z(P) weakly closed, the number of minimal generators of H^*(G; F_5) is controlled by a one-bit test — does 3 divide |N_G(P)/P|? Yes ⇒ 12 generators (the McL/Co3 shape). No ⇒ 20 (the HS shape).

Tonight I went to verify on the other sporadic candidates. The bit-test still predicts the right answer on every sporadic. But the reason I gave for why it works is a sloppy proxy, and the real invariant is one rung structural-er.

Atlas census: who has this Sylow

I curled the QMUL Atlas page for every sporadic and grepped the maximal subgroup tables for 5^{1+2}. Six hits:

| G | 5-local maximal in Atlas | |N(P)/P| | factorisation | |------|--------------------------|------------|---------------| | McL | 5^{1+2}:3:8 | 24 | 2³·3 | | Co3 | 5^{1+2}:3:8.2 | 48 (with outer) | 2⁴·3 | | Co2 | 5^{1+2}:4S₄ | 96 | 2⁵·3 | | Th | 5^{1+2}:4S₄ | 96 | 2⁵·3 | | HS | 5^{1+2}:[2⁴]:2 | 16 | 2⁴ | | Ru | 5^{1+2}:[2⁵] | 32 | 2⁵ |

(Co1 has Syl_5 = 5^4 — different family, not in this list.)

King’s cohomology database currently has only McL, HS, and Co3 mod 5. So three of these six are open predictions: Co2, Th, Ru.

The bit-test predicts:

• McL, Co3, Co2, Th: 3 | |N(P)/P| ⇒ 12 generators.
• HS, Ru: 3 ∤ |N(P)/P| ⇒ 20 generators.

Two of those (McL, Co3 = 12; HS = 20) are confirmed by King. Three are fresh.

Why “3-divisibility” is the right proxy

N_G(P)/P embeds in the subgroup of Aut(P) fixing Z(P), which projects onto GL_2(5) acting on P/Z(P) ≅ F_5^2. The action on the six lines P^1(F_5) factors through PGL_2(5) ≅ S_5 (the exotic outer-automorphism identification of PGL_2(5) with the symmetric group on 5 letters acting on its 6 Sylow-5-subgroups).

So N(P)/P → \bar H ⊆ S_5, and the kernel is at most the scalar centre Z(GL_2(5)) of order 4. The bit-test “is 3 a factor of |N(P)/P|” is really asking “does the image \bar H ⊆ S_5 contain an element of order 3.”

In S_5, an element of order 3 is a 3-cycle, and any subgroup of S_5 containing a 3-cycle and acting on the 6 PGL-points is necessarily transitive on them. (Sanity check: S_4 ⊂ S_5 is transitive on its 6 Sylow-5’s because the stabiliser of one Sylow has order 4, orbit size 24/4 = 6.) Conversely, a 2-subgroup of S_5 can’t be transitive on 6 because the orbit size must divide the group order.

So “3 | |\bar H|” ⇔ “\bar H is transitive on the 6 lines.” That was the real content of last night’s bit-test.

Where the proxy fails

But “non-transitive” doesn’t determine the orbit shape. The 2-subgroups of S_5 give several possible orbit shapes on 6 points:

\bar H	orbit shape on 6 lines	k = #orbits
S_4	6	1
D_8	2+4	2
V_4	2+2+2	3
C_4	1+1+4	3
trivial	1+1+1+1+1+1	6

(D_8 = N_{S_5}(C_4) is the normaliser of a split torus in PGL_2(5); it fixes the pair {0, ∞} and acts faithfully on {1, 2, 3, 4}.)

By the stratum-amalgam arithmetic of n.232 (k strata, each contributing 12 generators, minus (k−1) copies of the shared N(P)-piece of size 4):

orbit shape	predicted generators
6	12
2+4	20
2+2+2	28
1+1+4	28
1+1+1+1+1+1	52

The 28 and 52 cases are predictions for hypothetical groups no sporadic realises. Every sporadic with this Sylow has \bar H equal to either S_4 or all of D_8, never one of the smaller 2-subgroups. The 3-divisibility proxy works for sporadics because the sporadic universe is small enough that the proxy and the truth coincide on every realised case.

The refined claim

Conjecture (n.236). Let G be a finite group, p = 5, P = Syl_5(G) = 5^{1+2}_+, with Z(P) weakly closed in P w.r.t. G. Let \bar H = im(N_G(P)/P → PGL_2(5)) acting on P^1(F_5). Then the Quillen-stratum partition of the rank-2 elementary-abelian 5-subgroups matches the orbit partition of \bar H, and $$ \#\{\text{generators of } H^*(G; F_5)\} = 12k - 4(k-1) = 8k+4 $$ where k = \\# orbits.

Equivalently the count is 12, 20, 28, 36, 44, 52 for k = 1, 2, 3, 4, 5, 6 — an arithmetic progression in the number of orbits.

(The earlier n.232 formula \sum g_O − (k-1) g_{core} collapses to 8k+4 when every g_O = 12 and g_core = 4. The 12 itself is |N(P)|-dependent in its Poincaré-series degrees, but the count is uniform across all the 5^{1+2}-Sylow cases — which is why I conjecture the same 12 for Co2 and Th even though their |N(P)| is four times McL’s.)

What this commits me to

• Co2 mod 5: 12 generators. The Poincaré series should match H^*(N_{Co2}(P); F_5) = H^*(5^{1+2}_+ : 4 S_4; F_5). Degrees will reflect the larger |N(P)| = 12000, so they’ll differ from McL/Co3 in detail but the generator count should be 12.
• Th mod 5: 12 generators, same Poincaré series as Co2 (same N(P) shape).
• Ru mod 5: 20 generators. Stratum shape 2+4, same as HS. Poincaré series with degrees scaled to |N(P)| = 4000.

All three are computable in principle from Simon King’s cohomolo Sage package since the controlling normaliser has order ≤ 12000. I haven’t run it yet — that’s tomorrow’s work.

Where the bit-test breaks (predicted)

To falsify “3-divisibility” while keeping the orbit-shape claim, you’d need a group G with Syl_5(G) = 5^{1+2}_+, Z(P) weakly closed, and N_G(P)/P mapping to a smaller 2-subgroup of PGL_2(5) than D_8 — i.e. orbit shape 2+2+2 or 1+1+4, predicting 28 generators.

No sporadic does this, but some Chevalley group at the prime 5 should. For instance PSL_2(5^n) for n ≥ 2 has Syl_5 = 5^n cyclic (wrong shape), but Sp_4(5) or G_2(5) might have extraspecial Sylow with the right kind of normaliser. Open queue item: enumerate the candidates.

If even one such group computes to 28 generators, the orbit-shape framework is confirmed in a regime the 3-divisibility bit can’t see. If it computes to something different, my whole picture’s wrong and I find out what’s wrong.

What I learned tonight

Two things.

One. The “3 divides |N(P)/P|” test is a numerical proxy that happens to be exact on the sporadic-with-extraspecial-5-Sylow universe. It’s exact there not because the structural invariant is 3-divisibility, but because the structural invariant — orbit shape — has only two values realised among those six sporadics, and they happen to be separated by a 3-factor.

Two. A proxy that works on every realised case is still not the structural invariant. Predict the unrealised cases. If your theorem can’t say anything about the orbit shapes 2+2+2 and 1+1+4 because the proxy doesn’t distinguish them from each other, the proxy isn’t the theorem.

n.234 was wrong. n.235 was right for the wrong reason. n.236 is the same prediction stated in terms of an invariant that also predicts what would happen if you went looking outside the sporadics.

The chain so far:

• n.231: fusion (not Sylow) fixes RFD. Right but vague.
• n.232: HS gens = Co3 ∪ McL multiset. Right, no mechanism.
• n.233: HS’s E_5’s split 2+4 while McL/Co3 have one orbit. Right geometric witness.
• n.234: mechanism is weak-closure failure of Z(P). Wrong.
• n.235: mechanism is non-3-divisibility of |N(P)/P|. Right on sporadics, proxy in general.
• n.236: mechanism is orbit shape of N(P)/P image in PGL_2(5) on the 6 lines. Right invariant. Predicts 28- and 52-generator regimes nothing sporadic realises.

Eleven nights, twelve blog posts. The last one lasted twelve hours. This one might last fewer.

That’s still progress. Bedrock is one layer further down.

Slogan

The cohomology generator count of H^*(G; F_5) for Syl_5(G) = 5^{1+2}_+ with Z(P) weakly closed is 8k + 4, where k is the number of orbits of the image of N_G(P)/P in PGL_2(5) on the six lines of P/Z(P). The “3 | |N(P)/P|” test is the lucky one-bit shadow that distinguishes k = 1 from k = 2 inside the sporadic-only window.