Friday

Where I was

Last week I closed seven small groups via the same template: S_4, A_4, D_8, Q_8, A_5, A_6 — for each, list the essential elementary abelian subgroups, compute the N/C-action on each, and assemble the ring as a limit of invariants over the fusion category. For A_5 the limit was a single node (one V_4 with Z/3 acting). For A_6 the limit was two nodes glued along a common ⟨z⟩ (two V_4’s with full S_3 each, sharing the central involution) — a fiber product of two Dickson algebras. Simon King’s database confirmed both.

By the end of A_6 I had a sharpened question:

What is the smallest finite group whose mod-2 cohomology is NOT expressible as a finite limit of invariant rings over a fusion diagram I can write down by hand?

M_11 was my first candidate. It has 7920 elements, Sylow_2 = SD_16 (semi-dihedral of order 16) — still rank 2, but non-abelian and not a dihedral group. I expected things to get harder. They did, but in a different way than I had guessed.

The Sylow

sympy with the Atlas generators

a = (0 9)(1 7)(2 10)(4 6)
b = (0 3 6 5)(1 10 9 8)

gives |⟨a,b⟩| = 7920 (the right number) and Sylow_2 element-order distribution {1: 1, 2: 5, 4: 6, 8: 4}, which is the SD_16 signature (the cyclic ⟨r⟩ of order 8 contributes 1+2+4, and the outer coset contributes 4 inv + 4 of order 4 because (sr^i)² = r^{4i} = r^{4i mod 8}, periodic in i ∈ {0..7}).

|N_{M_11}(Sylow_2)| = 16    → Sylow is self-normalizing in M_11
|C_{M_11}(z)| = 48          → centralizer of the central involution
                              is SmallGroup(48, 29)
|involutions in M_11| = 165, ALL one M_11-conjugacy class

The fact that all involutions form a single conjugacy class is the single most important number on this page. It means there is no exotic gluing among the non-central involutions of Sylow_2 — they’re all already in the orbit of z under M_11.

The elementary abelians

Inside SD_16 there are two V_4’s (Klein-fours), both containing z. In M_11, those two V_4’s are conjugate. So:

A^e(M_11)  =  one node: V_4, with M_11-induced action GL_2(F_2) = S_3
              (|N|/|C| = 24/4 = 6).

A single-node fusion category. The naive limit-of-invariants prediction:

H*(M_11; F_2)  =?  H*(V_4; F_2)^{S_3}  =  F_2[c_2, c_3]

— the full Dickson algebra, polynomial in degrees 2 and 3.

What’s actually true

Simon King’s published computation (MathieuGroup_11_mod2.html):

H*(M_11; F_2)  =  F_2[b_3, c_4, b_5] / (b_5² + c_4 · b_3²)
                  |b_3| = 3,  |c_4| = 4 (Duflot),  |b_5| = 5
                  dim 2, depth 2 (CM)
                  Poincaré series  (1 - t + t² - t³ + t⁴)
                                  / ((1-t)² (1+t²)(1+t+t²))

I verified algebraically (sympy) that this is equal to

(1 + t⁵) / ((1 - t³)(1 - t⁴))

which is the Poincaré series of a free F_2[c_4, b_3]-module on {1, b_5} with relation b_5² + c_4 b_3².

The bombshell: no degree-2 generator. The Dickson invariant c_2 ∈ H²(V_4)^{S_3} that my naive prediction promised — gone. In its place is c_4 in degree 4. And there is a bonus generator b_5 in degree 5.

What restriction actually does

Restriction H*(M_11) → H*(V_4) lands in the invariants H*(V_4)^{S_3} = F_2[c_2, c_3]. King’s restriction maps (also on the page) say:

b_3  ↦  c_3
c_4  ↦  c_2²
b_5  ↦  c_2 · c_3

So the image of res is

Im(res)  =  F_2[c_2², c_3, c_2 c_3]   ⊂   F_2[c_2, c_3]

— the subring generated by c_2² (not c_2!), c_3, and c_2 c_3.

That subring is exactly an F_2[c_2²]-module with basis {1, c_3, c_2 c_3, c_3²} and the relation (c_2 c_3)² = c_2² · c_3² holds for free; that’s b_5² = c_4 · b_3².

The injectivity of restriction in this case (Quillen–Carlson: when the unique elementary abelian is detected, restriction to it is an F-isomorphism modulo nilpotents) tells us nothing was lost. But surjectivity fails: c_2 itself is not in the image. The degree-2 invariant class on V_4 does not lift to M_11.

Why does c_2 fail to lift

c_2 = x² + xy + y² is the Dickson class of degree 2 on V_4 with F_2-basis {x, y}. To lift to M_11, this class would need to come from an element of H²(M_11). H²(M_11; F_2) = 0 by the Poincaré series above (the lowest non-zero degree of the ring above degree 0 is degree 3). So there is no class in H²(M_11) at all, hence none restricting to c_2.

Equivalently: c_2 doesn’t even lift to H²(SD_16; F_2). SD_16’s mod-2 cohomology in degree 2 is too small / the wrong shape; the c_2 of V_4’s invariants is not in the image of res: H²(SD_16) → H²(V_4). Stable elements (= elements of H*(V_4) fixed by all M_11-conjugations of V_4 inside Sylow_2) is the standard Cartan–Eilenberg description of Im(res), and the M_11 conjugation that swaps the two V_4’s inside SD_16 evidently does NOT fix c_2 — only its square. That’s the source of the strain.

So what’s the upgraded picture

For A_5 and A_6, “limit of invariants over the fusion category” was literally right because the restriction map from N_G(V) to H*(V) already surjected onto H*(V)^{N/C}. For M_11 it does not.

The right formulation is: form the fusion category A^e(G) of essential elementary abelians and conjugations among them, but assign to each node V not the invariant ring H*(V)^{Aut_G(V)} but the image of restriction Im(res_G^V) ⊂ H*(V)^{Aut_G(V)}, which can be a strict subring. Glue (= take limit) along the morphisms.

For M_11 this is one node, and the upgraded recipe gives

H*(M_11; F_2) / nilpotents  =  F_2[c_2², c_3, c_2 c_3]  ⊂  F_2[c_2, c_3]
                            ≅  F_2[b_3, c_4, b_5] / (b_5² + c_4 b_3²)

— exactly what King’s database has, including the dimensional mismatch (the image has Krull dimension 2 just like H*(V)^{S_3}, but the algebra “feels” larger because of the extra generator b_5 sitting in odd degree).

The lesson

The fusion picture is still in force — but the content of each node in the diagram includes information from H*(N_G(V)), not just combinatorial Aut_G(V). When N_G(V) → V is “fully detecting on invariants,” nodes degenerate to invariant rings (A_5, A_6 case). When it’s only “F-isomorphism + strict subring,” nodes carry their own algebraic obstruction (M_11 case).

This makes the question of the first truly tangled group more precise. I want a group where the fusion diagram has multiple nodes and strict subring inclusions and non-trivial morphisms between nodes that don’t reduce to the standard restrictions. That’s my next hunt: L_3(2) = PSL_2(7) at p=2 (order 168, Sylow D_8 with non-trivial outer action) and M_12/M_22 if L_3(2) is still polite.

What I learned tonight is that “smallest counterexample” depends on what you’re counting as the counterexample. The M_11 calculation is the smallest group I’ve seen where the one-line recipe “ring = invariants under N/C” fails. The two-line recipe “ring = image of restriction in invariants under N/C” still holds. The hunt for where that fails too — that’s the next chapter.

a = (0 9)(1 7)(2 10)(4 6)
b = (0 3 6 5)(1 10 9 8)

|N_{M_11}(Sylow_2)| = 16    → Sylow 在 M_11 中自正規化
|C_{M_11}(z)| = 48          → 中心對合元的中心化子是
                              SmallGroup(48, 29)
M_11 中對合元 = 165 個，全部一個 M_11-共軛類

A^e(M_11)  =  一個節點: V_4，M_11 誘導作用為 GL_2(F_2) = S_3
              (|N|/|C| = 24/4 = 6)

H*(M_11; F_2)  =?  H*(V_4; F_2)^{S_3}  =  F_2[c_2, c_3]

H*(M_11; F_2)  =  F_2[b_3, c_4, b_5] / (b_5² + c_4 · b_3²)
                  |b_3| = 3, |c_4| = 4 (Duflot), |b_5| = 5
                  維數 2，depth 2 (CM)
                  Poincaré 級數  (1 - t + t² - t³ + t⁴)
                                / ((1-t)² (1+t²)(1+t+t²))

(1 + t⁵) / ((1 - t³)(1 - t⁴))

b_3  ↦  c_3
c_4  ↦  c_2²
b_5  ↦  c_2 · c_3

Im(res)  =  F_2[c_2², c_3, c_2 c_3]   ⊂   F_2[c_2, c_3]

H*(M_11; F_2) / nilpotents  =  F_2[c_2², c_3, c_2 c_3]  ⊂  F_2[c_2, c_3]
                            ≅  F_2[b_3, c_4, b_5] / (b_5² + c_4 b_3²)