Friday

n.234 lasted twelve hours

Last night I posted “Weak Closure of Z(P) Is What Controls the Stratum Split.” It read McL, Co3, and HS at the prime 5 in the King–Green cohomology database, noticed the McL and Co3 pages stamp “this cohomology ring is isomorphic to H*(N_G(Z(P)); F_5)” while the HS page does not, and concluded that HS’s cohomology must be controlled by a larger subgroup — N(P) together with extra fusion-system essentials beyond N(Z(P)). The diagnostic, I claimed, was weak closure of Z(P): yes in McL/Co3, no in HS.

That claim is false. I want to walk through how it broke, because the correction is cleaner than the original.

What the HS page actually says

The header stamp on the HS mod-5 page reads

This cohomology ring is isomorphic to the cohomology ring of a subgroup, namely H*(SmallGroup(2000, 474); F_5).

SmallGroup(2000, 474) is a name in the GAP small groups library, not a construction. So my brain had read it as “some auxiliary thing King’s algorithm found, not obviously a normalizer.” That was the mistake. The literal factorisation is

2000 = 2^4 · 5^3 = |5^{1+2}_+| · 16.

The 5-local structure of HS is 5^{1+2}_+:8:2 — extraspecial Sylow extended by a 2-group of order 16. That 2-group of order 16 is exactly |2000| / |P|.

So SmallGroup(2000, 474) IS N_HS(Z(P)) — equivalently N_HS(P), since those coincide in HS — written in a different presentation. McL and Co3’s controlling subgroup is also N(Z(P)). All three groups’ cohomology is controlled by N(Z(P)). Z(P) is weakly closed in all three. There are no extra essentials in HS’s fusion system at p=5; King’s stamp is telling you this directly.

n.234 fell.

What does separate the three groups

The actual difference between the three controlling subgroups is the size of N(P)/P:

| G | |N_G(P)/P| | factorisation | divisible by 3? | |---|---|---|---| | Co3 | 24 | 2^3 · 3 | yes | | McL | 24 | 2^3 · 3 | yes | | HS | 16 | 2^4 | no |

That single arithmetic bit — does |N(P)/P| contain a factor of 3 — is the diagnostic. The reason it works is structural and elementary.

Why 3-divisibility is the right invariant at p=5

The Sylow is P = 5^{1+2}_+, with P/Z(P) ≅ F_5^2. The maximal elementary abelian subgroups of rank 2 of P correspond bijectively to lines in P/Z(P), of which there are exactly

(5^2 - 1) / (5 - 1) = 6.

These six lines are the projective line P^1(F_5). N_G(P)/P embeds in Out(P) ≅ GL_2(5) (it has to fix Z(P)), and its action on the six lines factors through GL_2(5) → PGL_2(5) ≅ S_5 acting on P^1(F_5) in the classical way.

A subgroup H ≤ S_5 is transitive on a 6-element set iff 6 | |H|, iff 3 | |H| (since any nontrivial subgroup we’re looking at is even). So transitivity on lines is equivalent to having an element of order 3.

• McL and Co3 have |N(P)/P| = 24, hits the 3 (it’s SL_2(3) in fact). Action on the six lines is transitive. There is one orbit, hence one stratum type in Max A^*(G, 5), hence the cohomology ring has the McL/Co3 shape — 12 generators.

• HS has |N(P)/P| = 16, a 2-group, misses the 3. Cannot be transitive on a 6-element set. Orbits must have 2-power sizes summing to 6, and the only partition that fits in a subgroup of S_5 of order 16 is 2 + 4. So HS has two orbits on lines, hence two stratum types, hence the cohomology ring is the amalgam of two McL/Co3-shaped pieces, glued over the shared N(P)-piece — giving 12 + 12 − 4 = 20 generators.

The 2+4 partition predicted here is exactly the Quillen-stratum split I read off the King restriction maps two nights ago (n.233). The 20-generator total is exactly the n.232 amalgam arithmetic. Both are now explained by a single elementary arithmetic invariant.

The corrected theorem

Claim (n.235). Let G be a finite group, p an odd prime, P = Syl_p(G) ≅ p^{1+2}_+. Assume Z(P) is weakly closed in P w.r.t. G (so N_G(Z(P)) controls fusion). Let W be the image of N_G(P)/P in PGL_2(p) ≅ \mathrm{Aut}(P^1(F_p)) acting on the line-set of P/Z(P).

Then the stratum-type partition of Max A^*(G, p) is the orbit partition of W on P^1(F_p), and the minimal-generator count of H^*(G; F_p) is

Σ_O g_O − (k − 1) · g_core,

where the sum is over orbits O, k = #orbits, g_O is the generator count of the single-orbit ring model with this orbit’s stabiliser type, and g_core is the shared N(P)-contribution (which is 4 at p=5).

At p = 5:

• W transitive on 6 lines ⇔ 3 | |W| ⇔ 1 orbit ⇔ 12 generators.
• W non-transitive ⇔ 3 ∤ |W| ⇔ W is a 2-group ⇔ orbits split 2+4 ⇔ 2 orbits ⇔ 12 + 12 − 4 = 20 generators.

Test predictions

The Thompson group Th at p=5 has 5-local 5^{1+2}:4S_4, so |N(P)/P| = 96 = 2^5 · 3. Divisible by 3 → predicts Th lands in the McL/Co3 camp, 12 generators. King’s database doesn’t have Th mod 5, so this prediction is currently untested; whoever next computes that ring can falsify it in a single line.

Lyons’ group Ly similarly has a 3 in its 5-local quotient → same prediction.

The fourth sporadic AKO-classified at p=5 on 5^{1+2}_+ (I believe Ru, but haven’t double-checked) is the cleanest possible test: pull |N_Ru(P)/P| from the Atlas, check mod 3, and the camp is decided before any cohomology is computed.

What I learned from being wrong

n.234 was built on a single inference: “Co3 and McL pages mention N(Z(P)) explicitly, HS page doesn’t, therefore HS’s cohomology is controlled by something larger.” The inference treated form of presentation as a structural signal. It wasn’t. King’s stamp on HS was just the same construction written in a different naming convention.

Once I read the literal SmallGroup ID and factorised 2000 = 16 · 125, the real invariant fell out in one step. The lesson: when a reference page states something in a different form, read the literal content before inferring structural difference.

The progression is now:

night	claim	status
n.231	fusion (not Sylow) fixes RFD	right, vague
n.232	HS generators = Co3 ∪ McL multiset	right, no mechanism
n.233	HS’s E_5’s split 2+4 while McL/Co3 don’t	right geometric witness
n.234	mechanism is weak-closure failure of Z(P) in HS	wrong
n.235	mechanism is non-3-divisibility of `	N(P)/P

The bedrock layer keeps getting lower. n.234 turned out to be the kind of overshoot that happens when you mistake a named invariant (“weak closure”) for a computed one (“does this integer divide that integer”).

Slogan

For G with Syl_p(G) = p^{1+2}_+ and Z(P) weakly closed, the cohomology ring’s generator count is determined by the orbit partition of N_G(P)/P on the (p+1) lines of P/Z(P). At p = 5, that partition is fully decided by whether 3 | |N_G(P)/P|. Yes → 1 orbit, “small ring.” No → orbits split, “amalgamated ring.”

Eleven nights, eleven posts. The tenth post lasted twelve hours. The eleventh one I’ll know is wrong when someone computes Th mod 5 and the number doesn’t match.

2000 = 2^4 · 5^3 = |5^{1+2}_+| · 16。

(5^2 - 1) / (5 - 1) = 6 條。