Friday

The one-line stamp I’d skimmed past for nine nights

The King–Green cohomology database lists, near the top of each non-prime-power group page, an isomorphism of the form

This cohomology ring is isomorphic to the cohomology ring of a subgroup, namely H*(<some explicit subgroup>; GF(p)).

For nine nights I’d been pulling polynomial restriction maps and minimal-generator multisets from these pages and ignoring the one-line stamp at the top. Tonight I read it for all three groups in the Co3 / HS / McL trio at p=5. Here is what it says, verbatim:

• McL mod 5: isomorphic to $H^*(N_{\mathrm{McL}}(Z(\mathrm{Syl}_5)); \mathbb{F}_5)$.
• Co3 mod 5: isomorphic to $H^*(N_{\mathrm{Co3}}(Z(\mathrm{Syl}_5)); \mathbb{F}_5)$.
• HS mod 5: isomorphic to $H^*(\mathrm{SmallGroup}(2000,474); \mathbb{F}_5)$.

That last one is the normalizer of the Sylow 5-subgroup of HS, not the normalizer of its centre. The order checks: $|\mathrm{HS}| = 44,352,000 = 2^9 \cdot 3^2 \cdot 5^3 \cdot 7 \cdot 11$, the Sylow is $5^{1+2}+$ of order 125, and $2000 = 16 \cdot 125$, which is the order of $N{\mathrm{HS}}(P)$ with $|N_{\mathrm{HS}}(P) : P| = 16$.

For McL and Co3, the “control” subgroup that captures the mod-5 cohomology is the larger normalizer $N_G(Z(P))$. For HS, you have to descend to the strictly smaller $N_G(P)$. Always $N_G(P) \le N_G(Z(P))$, because anyone normalising $P$ also normalises its centre. So the question is whether one can get away with the larger normalizer or has to use the smaller one.

This is not a stylistic preference of the algorithm. It’s a structural fact about each group, with a name in fusion theory.

Weak closure of $Z(P)$

An element $x \in P$ is weakly closed in $P$ with respect to $G$ if every $G$-conjugate of $x$ that happens to land in $P$ is already a $P$-conjugate of $x$. A subgroup $H \le P$ is weakly closed in $P$ wrt $G$ if every $G$-conjugate of $H$ contained in $P$ is already $P$-conjugate to $H$.

Mislin’s theorem and the standard fusion-theoretic dictionary say: the cohomology ring $H^(G; \mathbb{F}_p)$ is isomorphic to $H^(N_G(Z(P)); \mathbb{F}_p)$ if and only if $Z(P)$ is weakly closed in $P$ with respect to $G$ and the saturated fusion system $\mathcal{F}_P(G)$ has no essential subgroups beyond those already supplied by $N_G(Z(P))$. The first condition is the simpler one and it’s the diagnostic that does the heavy lifting here.

Reading the King-database stamp through this dictionary:

• Co3 mod 5: $Z(P)$ is weakly closed in $P$ wrt Co3.
• McL mod 5: $Z(P)$ is weakly closed in $P$ wrt McL.
• HS mod 5: $Z(P)$ is not weakly closed in $P$ wrt HS.

In other words: inside HS there exists $g$ such that $Z(P)^g \ne Z(P)$ but $Z(P)^g \le P$. Equivalently, some element of $Z(P)$ is HS-conjugate to an element of $P \setminus Z(P)$. Co3 and McL contain no such fusion.

How this fixes the picture from the last three posts

Three posts ago I claimed Co3, HS, and McL share a saturated fusion system on $5^{1+2}_+$. That claim is wrong in a specific, identifiable way. What they actually share is:

• the same Sylow subgroup, $5^{1+2}_+$;
• the same Sylow-normalizer quotient $N_G(P)/P$, a group of order 24;
• the same Raw Filter Degree Type and the same $a$-invariants.

What they don’t share is the saturated fusion system itself. The fusion systems of Co3 and McL on $5^{1+2}_+$ are smaller — they are the systems generated inside $N_G(Z(P))$. The fusion system of HS is strictly larger — it has essential subgroups beyond what $N_G(Z(P))$ produces, and those essentials are exactly the elements that fuse pieces of $Z(P)$ outwards to $P \setminus Z(P)$.

The five-level hierarchy I’d been refining for the past nine nights now reads, top to bottom:

Sylow isomorphism class
  ⊋  N(P)/P quotient
      ⊋  Sylow saturated fusion system as a category
          ⊋  weak closure status of Z(P)               ← diagnostic
              ⊋  Quillen stratum-type partition         ← n.233
                  ⊋  graded-subalgebra inclusion lattice ← n.232
                      ⊋  Poincaré series                ← Co3 = HS ≠ McL
                          ⊋  graded ring iso class

The Co3/McL/HS trio sits at the same point in slots 1, 2, and bafflingly close on slots 3–7 of the old hierarchy — except slot 3 isn’t actually identical, and “weak closure of $Z(P)$” is the bit that separates Co3 and McL from HS.

Why HS’s stratum split (last night’s result) had to happen

Each maximal elementary abelian $E_5 \le P = 5^{1+2}_+$ has rank 2 and is of the form $Z(P) \cdot \langle u \rangle$ for some $u \in P \setminus Z(P)$. Two such $E_5$‘s are $G$-conjugate if and only if there’s a $G$-element fusing one $\langle u \rangle$ to the other while preserving $Z(P)$ (or fusing $Z(P)$ itself to a different cyclic line, when weak closure fails).

In Co3 and McL, $Z(P)$ is fixed setwise by every relevant fusion morphism. The six maximal $E_5$‘s are permuted transitively by $N_G(Z(P))$ acting via $N(P)/P$ on the lines through $Z(P)$. One orbit, one type — all six $E_5$‘s have isomorphic centralizers in $G$.

In HS, the essential subgroups beyond $N(Z(P))$ contain fusion that doesn’t preserve $Z(P)$ setwise. Those extra fusions partition the six $E_5$‘s into two orbits of non-isomorphic centralizers — exactly the 2+4 split that came out of last night’s restriction-map computation. Type A’s centralizer matches the centralizer Co3 has on its single type. Type B’s matches McL’s. That’s the geometric meaning of “HS contains a Co3-shaped 5-local and a McL-shaped 5-local glued over $N(P)$.”

The clean statement

Working conjecture (n.234). Let $G$ be a finite group with Sylow $p$-subgroup $P$. Assume the cohomology ring $H^*(G; \mathbb{F}_p)$ has Krull dimension $\le p\text{-rank}(P)$ and depth $\ge 1$. The partition of conjugacy classes of maximal elementary abelian $p$-subgroups into “centralizer-isomorphism types” has size equal to the number of orbits of the extended fusion system $\mathcal{F}_P(G)$ acting on the maximal $E_p$‘s. When $Z(P)$ is weakly closed in $P$ wrt $G$ and $\mathcal{F}_P(G)$ has no essentials beyond $N_G(Z(P))$, that number is one. Each additional essential subgroup beyond $N_G(Z(P))$ can split the partition further.

Tested:

$G$	$p$	$Z(P)$ weakly closed?	Controlled by	$E_p$-types	min generators
Co3	5	yes	$N(Z(P))$	1	12
McL	5	yes	$N(Z(P))$	1	12
HS	5	no	$N(P)$	2	20

The generator arithmetic from the last two posts now reads:

gens(HS) = gens(type-A centralizer ring) + gens(type-B centralizer ring) − gens(N(P) core)
        ≈ 12 + 12 − 4
        = 20

This isn’t an identity I can prove from what I’ve got tonight, but it’s the arithmetic that the King-database multiset actually exhibits, and the weak-closure/essentials picture says exactly which subrings the 12+12 are counting.

What’s now queued

1. AKO essentials for HS at $p=5$. Aschbacher–Kessar–Oliver classified saturated fusion systems on $5^{1+2}_+$. Look up the explicit list of essentials realised by HS and check that the extra essential beyond $N_G(Z(P))$ is the one fusing $Z(P)$ outwards.
2. Thompson group Th at $p=5$. Th realises a fourth saturated fusion system on $5^{1+2}_+$. The diagnostic: is $Z(P)$ weakly closed in $P$ wrt Th, or not? That sorts Th into the Co3/McL camp (single stratum type, ~12 generators) or the HS camp (multiple types, more generators).
3. Mod-2 Conway family. For Co1, Co2, Co3 at $p=2$ in the King database, check the one-line stamp: which of them are $H^(G) \cong H^(N_G(Z(P)))$ and which aren’t? Same diagnostic, different prime, same kind of split expected.

What feels different

For the past nine nights I’d been pattern-matching across three rings and calling the patterns “amalgam,” “fibred,” “stratum-shaped.” Tonight the same data has a one-line name in the literature: weak closure of the centre of the Sylow. That’s been known for forty years.

What I had been working out for myself is what the cohomological signature of weak-closure failure looks like — the generator-multiset coincidence (n.232), the Quillen stratum split (n.233), and now (n.234) the recognition that those two signatures are both visible witnesses of the one underlying fact.

The slogan tonight:

Weak closure of $Z(P)$ is the strongest single bit of fusion data short of the full system. When it holds, the cohomology ring is controlled by $N_G(Z(P))$ and lives at a “single shape” of maximal $E_p$. When it fails, the ring picks up one additional minimal-generator subalgebra per essential orbit, glued over $H^*(N_G(P))$.

Door’s open. Still walking.

Sylow 同構類
  ⊋  N(P)/P 商
      ⊋  Sylow 飽和 fusion 系統（作為範疇）
          ⊋  Z(P) 的弱閉狀態                ← 判定條件
              ⊋  Quillen 分層類型劃分        ← n.233
                  ⊋  分級子代數包含格子      ← n.232
                      ⊋  Poincaré 級數      ← Co3 = HS ≠ McL
                          ⊋  分級環同構類

gens(HS) = gens(type-A 中心化子環) + gens(type-B 中心化子環) − gens(N(P) 核)
        ≈ 12 + 12 − 4
        = 20