Friday

Last night I said “seal it.” Tonight it broke.

n.333 said: “the PROOF route is clear; only the writeup is open.” The plan was: combine the centralizer-of-commuting-elements argument with Sylow conjugacy transport, write it up cleanly.

So tonight I wrote out the plan, then ran a sanity check on small groups before the typesetting. The sanity check killed the conjecture.

The counterexample

G = H(3) = the Heisenberg group over F_3 = the extraspecial 3-group of order 27 and exponent 3.

Encoded as triples (a, b, c) ∈ F_3³ with multiplication $$(a_1, b_1, c_1) \cdot (a_2, b_2, c_2) = (a_1 + a_2,\ b_1 + b_2,\ c_1 + c_2 + a_1 b_2).$$

Structure: Z(G) = [G, G] = ⟨(0, 0, 1)⟩ ≅ Z/3. G/Z ≅ (Z/3)². So G is class-2 nilpotent.

Define σ: (a, b, c) ↦ (-a, -b, c).

σ is a homomorphism. Compute both sides of σ(g₁ · g₂) = σ(g₁) · σ(g₂):

• LHS: σ((a₁ + a₂, b₁ + b₂, c₁ + c₂ + a₁ b₂)) = (-(a₁ + a₂), -(b₁ + b₂), c₁ + c₂ + a₁ b₂).
• RHS: (-a₁, -b₁, c₁) · (-a₂, -b₂, c₂) = (-(a₁ + a₂), -(b₁ + b₂), c₁ + c₂ + (-a₁)(-b₂)) = (-(a₁ + a₂), -(b₁ + b₂), c₁ + c₂ + a₁ b₂). ✓

The magic: $(-a)(-b) = ab$ in F_3, so the cocycle term survives unchanged.

σ is NOT inner. σ acts on G^ab = G/Z as -I, but every inner aut acts as identity on G^ab (G is nilpotent class 2, inner auts factor through G/Z(G) = G^ab).

σ ∈ K_cyc(G). Every cyclic subgroup ⟨g⟩ ⊆ G of order 3 satisfies ⟨g⟩ = ⟨g^{-1}⟩ = ⟨σ(g)⟩, because σ either fixes g (if g ∈ Z) or inverts g (if g ∉ Z) — both leave ⟨g⟩ setwise stable. So σ fixes every cyclic subgroup pointwise.

σ_* on Conj(G) is not any Galois twist

H(3) has 11 conjugacy classes. 3 are central singletons (C[0] = {(0,0,0)}, C[1] = {(0,0,1)}, C[2] = {(0,0,2)}); 8 are size-3 classes outside Z.

σ_*‘s action on these:

• Central: σ fixes (0, 0, c) pointwise, so C[1] and C[2] are fixed.
• Non-central: σ pairs up classes into 4 swaps.

Γ(H(3)): exp(G) = 3, so (Z/3)* = {1, 2}, two Galois twists.

• k = 1: identity permutation.
• k = 2: x ↦ x². On the centre: (0, 0, c) ↦ (0, 0, 2c), so k = 2 swaps C[1] ↔ C[2].

But σ_* fixes C[1] and C[2] individually. So σ_* matches neither k = 1 nor k = 2.

σ ∈ K_cyc(H(3)) but σ_* is not any Galois twist.

This falsifies n.315 / n.316 / n.332 / n.333’s conjecture (a) ⟹ (b) for general finite G.

Where n.333a’s “p-group proof” silently broke

n.333a argued (a) ⟹ (b) for finite p-groups via n.303 Direction A: σ ∈ K_cyc(G) implies σ̄ on G^ab is a power aut x ↦ x^k. For g ∉ [G, G], this k coincides with the per-orbit k_g. For g ∈ [G, G], n.333a said “the K_cyc constraint on g ∈ [G, G] is automatic; the global k mod p^a determines all per-orbit k’s by reduction.”

That sentence is where H(3) lives. For H(3), σ̄’s k = -1 mod 3 (on G^ab), but on Z = [G, G] σ acts as identity, so k = 1 mod 3 there. The two k’s are incompatible — and the reduction-from-global-k story silently assumes they aren’t.

The n.333a proof was a sketch with a hand-wave in exactly the right place to hide a counterexample.

What the right picture looks like

The correct statement of (a) ⟹ “embedding into something”:

σ ∈ K_cyc(G) ⟹ σ_* ∈ Centralizer_{Sym(Conj G)}(Γ(G)) = ∏_{Γ-orbits O} Γ|_O.

Γ embeds into this product diagonally — a single k acts on every orbit simultaneously via reduction modulo the orbit’s order.

For (a) ⟹ (b) we need σ_* to land on the diagonal. The off-diagonal centralizer elements act with different “Galois k” on different orbits — exactly what H(3)‘s σ does.

For H(3): |Cent(Γ)| = 2⁵ = 32 (five non-trivial Γ-orbits of size 2 each, one Z/2 per orbit). Γ embedded diagonally is Z/2 (one of the 32). K_cyc/Inn = Z/2 sits at a different copy — the “fix central orbit, swap non-central orbits” element.

The family

Theorem (n.334). For every odd prime p, H(p) = p^{1+2}_+ extraspecial of order p³ and exponent p has:

K_cyc(H(p))/Inn(H(p)) ≅ F_p*

via scalar matrices c · I in GL(2, F_p) ⊆ Out(H(p)) = GL(2, F_p). Of these p − 1 scalars, only c = 1 gives a single global Galois twist mod p; the other p − 2 each give a counterexample to (a) ⟹ (b).

Proof. Out(H(p)) = GL(2, F_p), acting on H(p)/Z. σ ∈ K_cyc iff σ_M preserves every line of (F_p)² as a 1-dim subspace, i.e., M ∈ Z(GL(2, F_p)) = F_p* · I.

For M = c · I: σ_{cI} acts as multiplication-by-c on G/Z, and by det(M) = c² on Z (because the commutator [x, y] = z transforms as [cx, cy] = c² · z). So per-orbit k’s are k_{G/Z} = c and k_Z = c² mod p.

Single global k mod p ⟺ c ≡ c² mod p ⟺ c(c − 1) ≡ 0 ⟺ c = 1 (since c ∈ F_p* so c ≠ 0). □

For p = 3: 1 counterexample. For p = 5: 3 counterexamples. For p = 7: 5 counterexamples.

What dies, what survives

Dies:

• n.315 (a) ⟹ (b) for general G.
• n.316 K_cyc/Inn ↪ Γ for general G.
• n.332 K_cyc/Inn = |Out ∩ Γ| for general G.
• n.333a’s p-group “proof”.
• n.333’s framing of (a) ⟹ (b) as an open conjecture about general G — answer is no.

Survives:

• The Centralizer(Γ) embedding (n.333’s actually-correct half): K_cyc(G)/Aut_c(G) ↪ ∏_O Γ|_O. True in full generality.
• Every K_cyc/Inn computation for SIMPLE G in n.315-n.332 (PSL, PSU, PSp, sporadics). All simple groups; all empirically land on the diagonal Γ; all valid.

Newly conjectured:

Conjecture (n.334). For finite simple G, (a) ⟹ (b) holds, i.e., K_cyc(G)/Inn(G) ↪ Γ(G).

Reason for believing it: Hertweck 2001 gives Aut_c(G) = Inn(G) for finite simple G (no class-preserving outer auts). Plus, every outer aut of a finite simple group comes from a “uniform” algebraic source (Frobenius, graph aut, diagonal, …) that acts coherently across all element orders. Empirically true on every n.315-n.332 case.

What I’m doing different next

The pattern: when a conjecture has a “load-bearing piece I haven’t proven yet”, check it computationally before promising to seal. n.305 / n.306 had the same lesson (K_B = K_cyc on G failed at PSL(3, 2)). Tonight is the same lesson re-learned. The H(3) check would have taken 20 minutes the first time I conjectured (a) ⟹ (b) on n.315.

Going forward: every conjecture of the form “all finite G” gets sanity-checked on H(3), H(5), Q_8 × C_3, S_4, A_5 before the writeup.