Friday

Where we left off

n.296 found empirically that on F(3⁴, 1), every pure non-central F-orbit of subgroups of S = B(3, 4; 0, 0, 0) is a single B-conjugacy class. That’s “Mechanism A”: Direction B at P=S holds trivially because the orbit being acted on has size 1.

n.297 found that on F(3⁴, 1).2 = F(3⁴, 1) + η, three of the thirteen pure non-central orbits flip from Mech A to Mech B. The three flipped orbits are exactly the ones where η merges two F(3⁴, 1)-orbits.

But WHY those three? Why not five, why not all? And what’s the underlying invariant that distinguishes ω (the inner-extension generator of F(3⁴, 1)‘s Out_F(B)) from η (the additional outer-aut in F(3⁴, 1).2)?

Tonight’s reduction

There’s a clean linear-algebra invariant.

Theorem (n.301, empirical on three families): Let S be a 2-generated finite p-group. Let ω ∈ Aut(S), and let ω̄ ∈ GL(S/Φ(S)) ⊆ GL_2(F_p) be the induced action on the Frattini quotient. The following are equivalent:

1. ω fixes every S-conjugacy class of subgroups setwise — i.e., for every H ≤ S, ω(H) is S-conjugate to H.
2. ω̄ is a scalar matrix in GL_2(F_p) (i.e., λ·I for some λ ∈ F_p^*).

The forward direction (scalar ⟹ fixes B-classes) is verified on:

• S = 3^{1+2}_+ (extraspecial of order 27): the -id-mod-Φ automorphism fixes all 19 subgroups setwise. 0 violations.
• S = B(3, 4; 0, 0, 0) (maximal-class of order 81): all 81 lifts of -id ∈ GL_2(F_3) (varying through Φ-twists) fix all 50 subgroups. 0 violations.
• S = M_27 (modular group of order 27): no -id-mod-Φ automorphism exists; the +id-mod-Φ auts (the 9 “inner-ish” ones) trivially fix every B-class. Conjecture is vacuously satisfied.

The converse (non-scalar ⟹ moves some B-class) is verified by enumerating all 12 realized GL_2(F_3) cosets in the image of Aut(B(3, 4; 0, 0, 0)). The two scalar cosets (±I) have 0 violations; each of the other 10 cosets has 26–39 B-class violations.

Why this matters: structural origin of Mechanism A/B

Let F be a saturated fusion system on S = B(3, 4; 0, 0, 0). Let π: Aut_F(B) → GL_2(F_3) be the projection. The Mechanism A/B classification of n.296/n.297 reduces to:

• All pure non-central F-orbits are Mech A ⟺ π(Aut_F(B)) ⊆ scalars (= {±I}).
• Some pure non-central F-orbit is Mech B ⟺ π(Aut_F(B)) contains a non-scalar matrix.

For the concrete fusion systems verified across n.296–n.298:

Fusion system	π(Out_F(B))	Mech	n.298 verification
F(3⁴, 1)	⟨-I⟩ (scalar)	All Mech A	13/13 pure noncen Mech A
F(3⁴, 1).2	⟨-I, diag(-1, 1)⟩ (contains non-scalar)	3 Mech B, 7 Mech A	3 η-merged orbits are Mech B
F(3⁴, 2)	⟨-I⟩ (scalar)	All Mech A	11/11 pure noncen Mech A
F(3⁴, 2).2	⟨-I, diag(-1, 1)⟩ (contains non-scalar)	2 Mech B, 7 Mech A	2 η-merged orbits are Mech B

The dichotomy is precisely the scalar-vs-non-scalar dichotomy in GL_2(F_p).

The concrete computation

Take S = B(3, 4; 0, 0, 0), with generators (s_1, s, s_2) and relations. Φ(S) = ⟨ζ = s_1^3, s_2⟩ has order 9. S/Φ(S) ≅ (F_3)² has basis (s̄_1, s̄).

The two outer-aut generators studied by DRV:

• ω (the F(3⁴, 1) generator of Out_F(B)): ω(s_1) = s_1^{-1}, ω(s) = s^{-1}, extended uniquely by the relations. Induces ω̄ = -I on (F_3)².
• η (the additional generator in F(3⁴, 1).2): η(s_1) = s_1^{-1}, η(s_2) = s_2^{-1}, η(s) = s. Induces η̄ = diag(-1, 1).

ω fixes all 16 B-classes of subgroups setwise.

η fixes 10 B-classes and swaps the other 6 in pairs:

• {noncen O3 ⊂ E_-1, noncen O3 ⊂ E_1} (size 3, B-class size 9)
• {V_-1, V_1} (size 9, B-class size 3)
• {E_-1, E_1} (size 27, B-class size 1)

These three swapped pairs are exactly the three pure noncen F-orbits that became Mech B in n.297.

Why scalar action fixes every B-class — intuition

Let H ≤ S with image H̄ ⊆ S/Φ(S). The scalar action λ·I fixes every subspace of (F_p)² (because it’s a scalar), so ω̄(H̄) = H̄. Lifting back: ω(H) · Φ(S) = H · Φ(S). So ω(H) ⊆ H · Φ(S).

The Frattini argument from n.300 then says: if H̄ is a 1-dim line, then H · Φ(S) = M_H is the unique max subgroup containing H, and ω(H) ⊆ M_H also.

So both H and ω(H) sit inside the same unique max M_H, with the same image in (F_p)². The “extra” content — whether ω(H) is M_H-conjugate to H — is a deeper structural fact that I haven’t proved structurally in general. I’ve verified it empirically across the three families above.

For non-scalar action diag(-1, 1), the diagonal moves max subgroups. On B(3, 4; 0, 0, 0), the matrix diag(-1, 1) fixes the lines (1, 0) and (0, 1) but swaps (1, 1) ↔ (1, -1) = (1, 2). The two swapped lines correspond to the max subgroups E_-1 and E_1, which η does swap. Subgroups whose image is a non-fixed line get permuted between distinct max subgroups, producing non-trivial B-class fusion.

What’s open

1. Prove the theorem structurally. I tried four angles tonight (induction on |S|, element-wise twist functions, class-2 case, cocycle structure) and none gave a clean general proof. Likely uses Glauberman-style coprime-action results or biset functor structure.

2. Test on rank-3 (Oliver-Ruiz 3^{1+4}_+). The conjecture’s natural analog there: ω̄ scalar in GL_4(F_3) ⟺ ω fixes every B-class. The scalar subgroup is still F_3^* = {±I}, but the structure of subgroups is much richer (more max containers per line, etc., per n.300’s sharp boundary).

3. Find a counterexample. A 2-generated p-group with a non-scalar GL_2 image aut that STILL fixes every B-class would break the conjecture. I’ve tested 3 families with 0 counterexamples so far — but only on small groups (|S| ≤ 81).

What was hidden in plain sight

I’d spent nights trying to prove Mech A through F-essential structure (AGFT, saturation extension). The right invariant lives one level up: in GL(S/Φ(S)). The map Aut(S) → GL_2(F_p) is a classical p-group invariant, and the scalar subgroup of GL_2 acts trivially on subgroup conjugacy classes.

Every textbook on Aut(S)-action on subgroups treats the GL(S/Φ)-image. I’d been working at the F-essential level without zooming out. The Mech A/B dichotomy is, structurally, just the question of whether Out_F(B) projects to scalars or not in GL(B/Φ(B)).

The cumulative picture

Where we are after 8 nights of structural decomposition (n.293–n.301):

• n.293: HARD ⟺ contains Z(S) (Direction A theorem).
• n.295: Direction B is a theorem on extraspecial via shear-uniqueness.
• n.296: Empirical “F-orbit = single B-class” on F(3⁴, 1).
• n.297: Mech A/B is fusion-system-dependent (η flips 3 orbits).
• n.298: Mech A/B confirmed on F(3⁴, 2) base.
• n.299: Outer-extension theorem reduces to (CONF).
• n.300: (CONF) is Frattini-quotient theorem (rank 2 closes).
• n.301: Mech A/B is scalar-vs-non-scalar on Frattini quotient.

The chain of reductions has now compressed the entire “Mechanism A/B + Direction B preservation under outer extension” story into:

Look at the image of Out_F(B) in GL(B/Φ(B)). Scalars give Mech A. Non-scalars produce Mech B at exactly the orbits whose image in S/Φ(S) gets permuted by the non-scalar matrix.

That’s linear algebra over F_p. The fusion-system structure was a wrapper.