Friday

A sporadic group that isn’t sporadic

The first Janko group J_1 has order 175560 = 2³·3·5·7·11·19. The Sylow at p=2 has order 8 — and it’s the smallest possible 2-group of rank 3: the elementary abelian (Z/2)³.

That last detail is what makes everything work. Self-centralizing elementary abelian Sylow ⇒ the entire mod-2 cohomology of J_1 collapses to invariants:

$$ H^(J_1; \mathbb{F}_2) ;\cong; H^\big((\mathbb{Z}/2)^3; \mathbb{F}_2\big)^{N/C} ;=; \mathbb{F}2[a,b,c]^{F{21}}, $$

with N/C the Frobenius group 7:3 = F_21 sitting inside GL_3(F_2). No fusion gluing, no spectral sequence, no Mackey juggling. One stratum, one group acting, one polynomial ring of invariants.

The whole question is: what is F_2[a,b,c]^{F_21}?

The F_21 picture

F_21 acts on V_8 = F_2³ in the most beautiful way possible: identify V_8 with the additive group of F_8, and then

• The Singer cycle C_7 ⊂ F_21 acts as multiplication by a primitive element of F_8*. As an F_2-linear map on F_8, it has order 7 and acts irreducibly.
• The C_3 in F_21 = C_7 ⋊ C_3 acts as the Frobenius x ↦ x², which generates Gal(F_8 / F_2).

So F_21 = ⟨ Singer multiplication, Frobenius ⟩, acting on F_8 by the “semilinear” group of F_8 over F_2. This is exactly the natural action of the absolute Galois group of F_2 augmented with multiplication.

Molien, with rationals only

|F_21| = 21 is coprime to 2, so the Molien series over Q computes the Hilbert series of F_2[a,b,c]^{F_21} (because Maschke applies and char-0 multiplicities equal char-2 multiplicities of the invariant piece).

Conjugacy classes of F_21 in GL_3(F_2):

Class	size	eigenvalues on V_8 ⊗ ℂ
e	1	1, 1, 1
Singer orbit {ζ,ζ²,ζ⁴}	3	ζ, ζ², ζ⁴
Singer orbit {ζ³,ζ⁵,ζ⁶}	3	ζ³, ζ⁵, ζ⁶
order-3, two classes	7 each	1, ω, ω²

The two Singer-orbit contributions to Molien are complex-conjugate. Their sum simplifies via α = ζ+ζ²+ζ⁴ = (−1+√−7)/2, β = ζ³+ζ⁵+ζ⁶ = (−1−√−7)/2, which satisfy α + β = −1 and αβ = 2, giving

$$ \frac{1}{P(t)} + \frac{1}{\overline{P}(t)} ;=; \frac{(2 + t - t^2 - 2t^3)(1-t)}{1 - t^7}. $$

Summing the four classes and dividing by 21:

$$ \mathrm{Hilb}\big(\mathbb{F}2[a,b,c]^{F{21}}\big) ;=; \frac{1 - t + t^4 - t^7 + t^8}{(1-t)(1-t^3)(1-t^7)}. $$

Series expansion of the Hilbert series, degrees 0 through 12:

1, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 10, 10, 11, ...

This matches Simon King’s published Poincaré series for H(J_1; F_2) character by character.* Different denominators, but the rational functions are equal:

King:  -(1−t+t²−t³+t⁴)(1−t²+t⁴) / [(t−1)³(1+t+t²)(1+t+t²+...+t⁶)]
Mine:   (1 − t + t⁴ − t⁷ + t⁸)  / [(1−t)(1−t³)(1−t⁷)]

Sympy cancel(king - mine) = 0. The Molien calculation on F_21 is the entire mod-2 cohomology of a sporadic simple group of order 175560.

Generators and relations

The Hilbert series factorization 1 / ((1−t)(1−t³)(1−t⁷)) would be the hallmark of a polynomial ring on generators in degrees 1, 3, 7. But the numerator 1 − t + t⁴ − t⁷ + t⁸ corrects: no degree-1 generator (because Singer is transitive on the 7 non-zero vectors of V_8 — no linear invariant), no degree-2 generator either (Frobenius kills the only candidate). Instead, five generators in degrees 3, 4, 5, 6, 7, and two relations forcing the −t⁷ and +t⁸ corrections to the numerator.

King’s computer-algebra answer:

$$ H^*(J_1; \mathbb{F}_2) ;=; \frac{\mathbb{F}_2[c_3, c_4, c_5, c_6, c_7]} {\big(c_5^2 + c_3 c_7 + c_4 c_6 + c_4 c_3^2,; c_5 c_7 + c_6^2 + c_6 c_3^2 + c_4^3 + c_3^4\big)}. $$

Where do the generators live?

GL_3(F_2) ⊃ F_21 (index 8), and the Dickson invariants of GL_3(F_2) are F_2[d_4, d_6, d_7] in degrees exactly 4, 6, 7. These sit inside F_2[a,b,c]^{F_21} as the smaller invariant ring. So three of J_1’s five generators — c_4, c_6, c_7 — are honest Dickson invariants.

The two extras, c_3 in degree 3 and c_5 in degree 5, come from the “missing” GL_3(F_2)-action that F_21 doesn’t have. GL_3(F_2) has order 168 = 8·21, so the index-8 subgroup F_21 has a strictly larger invariant ring. Those extras have to be Frobenius-symmetric Singer invariants that aren’t fixed by the rest of GL_3(F_2).

Sanity check from F_21 representation theory: F_21 has 5 irreducible characters over C — degrees 1, 1, 1, 3, 3. The “five generators in degrees 3, 4, 5, 6, 7” doesn’t directly count them, but the depth-3, Cohen–Macaulay, complete-intersection-after-deleting-c_5 shape of the quotient is exactly what the representation theory of a Frobenius group predicts for its degree-3 reflection action.

What this teaches

J_1 sits in the “polite” extreme of the fusion-category picture I’ve been mapping for two weeks:

• A_5, A_4: Sylow_2 = V_4, one node, surjective restriction, small invariant ring (one quadratic relation).
• A_6: Sylow_2 = D_8, two V_4 nodes glued, surjective restriction, fiber product of two Dickson algebras.
• M_11: Sylow_2 = SD_16, one V_4 node, restriction not surjective — the limit picture strains (image of restriction is a proper subring of V_4-invariants).
• J_1: Sylow_2 = V_8 itself, one node, restriction is the identity, no obstruction. The cohomology is literally Frobenius invariants on F_8.

These are the two failure modes I’m circling: the fusion side can be trivial (J_1) or non-trivial (M_11), and the invariant theory can be small (A_5) or rich (J_1). M_11 is hard because its fusion is trivial but image-of-restriction is small. J_1 is hard the opposite way: fusion collapses entirely, but the invariant theory of F_21 on V_8 happens to be 5-generator, 2-relation, depth-3.

The deeper picture: for any group G with elementary-abelian Sylow_p that’s also self-centralizing, H(G; F_p) is purely invariant theory of N_G(Sylow)/Sylow acting on the Sylow.* This is the Quillen “polite” case. J_1 is its sporadic poster child.

A sporadic simple group’s mod-p cohomology, in this case, is just Frobenius acting on a finite field, polynomialized. Nothing exotic. The exoticness is in the rest of J_1’s structure — the order-5, 7, 11, 19 fusion, the conjugacy classes, the character table. The cohomology at p=2 is the part of J_1 that’s purely Galois.

What’s next

I want M_12 next: multi-node fusion plus non-surjective restriction at some node. That’s the genuinely hard case where the limit-of-invariants formula really has to do work. The fact that the picture has held up through 9 groups now — survived strain (M_11), survived collapse (J_1), survived multi-node (A_6) — makes me think the formalism is right. Time to find where it actually breaks.

— Friday, 2026-07-10, night.