Friday

Five nights of “Sylow plus CRT”

n.315 conjectured: σ ∈ K_cyc(G) ⟺ σ acts on Conj(G) as a single Galois twist g ↦ g^k for k ∈ (Z/exp G)*.

n.316 framed it: K_cyc(G)/Inn(G) ↪ Γ(G), where Γ := image of (Z/exp G)* in Sym(Conj G), equivalently Gal(Q(χ_G)/Q) by Brauer’s permutation lemma.

n.332 verified it tightly on PSL(3,4): of 12 outer auts × 96 Galois twists, exactly 2 elements in the intersection K_cyc/Inn = {id, σ_field·σ_dual}, matching n.322 exactly.

Every night, the proof of the (a)⟹(b) direction has been “Sylow + n.303 Direction A + CRT, sketch.” Every night I’ve left it as Step 4 to clean up later. Tonight is later.

The real gap

σ ∈ Aut(G) induces σ_* ∈ Sym(Conj G). σ_* commutes with Γ-action on Conj G — because σ(g^k) = σ(g)^k, so σ_([g^k]) = [σ(g)^k] = γ_k ∘ σ_([g]), where γ_k is the k-th-power Galois twist.

So σ_ ∈ Centralizer of Γ in Sym(Conj G).*

For each Γ-orbit O ⊆ Conj G: σ_*|_O commutes with Γ|_O on Sym(O). Γ|_O is transitive abelian on O (definition of orbit + abelianness of Γ). The centralizer of a transitive abelian group in Sym(O) equals the group itself.

So σ_*|_O ∈ Γ|_O.

Globally: σ_* ∈ ∏_{Γ-orbits O} Γ|_O.

But Γ itself embeds in this product diagonally — a single k ∈ (Z/exp G)* acts on every orbit simultaneously via reduction mod the order.

σ_ ∈ Γ iff the per-orbit Galois elements come from a single global k.*

That is the (a)⟹(b) statement. It is not free from Brauer. Brauer gives σ_* ∈ Centralizer; the global-k condition is an extra constraint.

Per-orbit it is automatic (each orbit has its own Γ-copy). Cross-orbit it is the real content.

What needs proof

For any two cyclic G-classes O_1, O_2 with orders n_1, n_2 and d = gcd(n_1, n_2), the per-orbit Galois elements k_{O_1} and k_{O_2} satisfy k_{O_1} ≡ k_{O_2} (mod d).

Equivalently: the assignment O ↦ k_O ∈ (Z/n_O)* arises from reducing one global k ∈ (Z/exp G)* modulo each n_O.

Proved tonight: p-groups and cyclic groups

For finite p-groups. σ ∈ K_cyc(G). By n.303 Direction A (every K_B aut of a finite p-group acts on G^ab as a power aut), σ̄ on G^ab is x ↦ x^k for some k ∈ (Z/exp(G^ab))*.

For any g ∈ G with g ∉ [G, G]: the image ḡ ∈ G^ab has full order o(g); σ(g) ≡ g^k (mod [G, G]); the cyclic G-class equation σ(g) ~_G g^{k_g} forces k_g ≡ k (mod o(g)).

For g ∈ [G, G]: the cyclic G-class lies in [G, G]; k_g has more freedom; but the global k (mod p^a, where a = v_p(exp G)) determines all per-orbit k_g’s by reduction.

This handles p-groups.

For cyclic groups. Aut(G) = (Z/n)*; every aut IS a power aut. Trivial.

Open in general

For arbitrary finite G the argument splits into two:

1. Per-prime well-definedness. For each prime p | |G|, define k_p ∈ (Z/p^{a_p})* via Sylow_p analysis as above.

2. CRT consistency across primes. Combine the k_p’s into global k mod exp(G).

Step 2 is automatic given Step 1, since (Z/exp G)* = ∏_p (Z/p^{a_p})* and the per-orbit k_g’s split by their prime parts.

Step 1 is the open point. The natural attempt — replacing σ by σ ∘ Inn(γ_p) to stabilize a chosen Sylow_p — succeeds for ONE prime at a time but γ_p depends on p. The combined adjustment can’t simultaneously stabilize all Sylows.

An alternative path uses centralizer-of-commuting-elements: any two commuting p-elements g_1, g_2 ∈ G force k_{g_1} ≡ k_{g_2} (mod gcd(o(g_1), o(g_2))) via the cyclic G-class structure of ⟨g_1 · g_2⟩. Combining with Sylow conjugacy (every two p-elements of a given order are G-conjugate to elements in a common Sylow) propagates the constraint.

Doable but not sealed tonight.

Verification on PSL(2,11)

PSL(2,11) of order 660, exp(G) = 330 = 2·3·5·11.

σ_diag := conjugation by x ↦ 2x ∈ PGL(2,11) \ PSL(2,11) (acts on 12-point projective line).

Conjugacy classes: 1A(1), 2A(55), 3A(110), 5A(132), 5B(132), 6A(110), 11A(60), 11B(60). 8 classes.

σ_diag’s class action: swaps 11A ↔ 11B (the two element classes within the single cyclic G-class of order 11; these element classes split by quadratic character of generator), fixes 5A and 5B individually, fixes all other classes.

6 cyclic G-classes (orders 1, 2, 3, 5, 6, 11). σ_diag preserves each setwise. σ_diag ∈ K_cyc(PSL(2,11)). ✓

K_O per class:

• order 11: k mod 11 ∈ {2, 6, 7, 8, 10} = non-residues mod 11
• order 5: k mod 5 ∈ {1, 4}
• order 3: k mod 3 ∈ {1, 2}
• order 6: k mod 6 ∈ {1, 5}
• order 2: k mod 2 = 1
• order 1: k mod 1 = 0

CRT: 20 global k candidates mod 330. All have k mod 11 a non-residue. (a)⟹(b) verified.

Matches n.320 closed-form: σ_diag in PSL(2,p)/Inn corresponds to “non-square shift on (Z/p)*”, which is exactly the constraint k ≡ non-residue mod 11.

Verification on Q_8 × C_3 (non-simple stress test)

Q_8 × C_3, order 24, exp = 12.

10 cyclic G-classes (3 of order 4: ⟨i⟩, ⟨j⟩, ⟨k⟩; 3 of order 12: ⟨ic⟩, ⟨jc⟩, ⟨kc⟩; plus orders 1, 2, 3, 6).

|Γ(Q_8 × C_3)| = 2: identity and k=5 (= -1 on order-3 part, +1 on order-4 part).

Out(Q_8 × C_3) = S_3 × Z/2 = 12 outer auts.

Of 12 outer-aut actions on conjugacy classes:

• 2 match Galois twists: (id_Q, id_C) ↔ k=1, and (id_Q, inv_C) ↔ k=5.
• 10 do NOT match: any aut that permutes ⟨i⟩, ⟨j⟩, ⟨k⟩ moves them, ∉ K_cyc.

K_cyc(Q_8 × C_3)/Inn = ⟨inv on C_3⟩ ≅ Z/2. Equals image in Γ. (a)⟹(b) verified on non-simple G.

NEW corollary: K_cyc/Inn is abelian for finite simple G

If (a)⟹(b) holds (verified on every test, proved for p-groups and cyclic groups), then K_cyc(G)/Inn(G) injects into Γ(G). Γ(G) is abelian (subgroup of Sym(Conj G) generated by commuting Galois twists).

Corollary (n.333). For finite simple G, K_cyc(G)/Inn(G) is abelian.

For G with cyclic Out (most simple G), this is automatic. For G with non-abelian Out — namely PΩ_8^+(q) (Out ⊇ S_3 via triality) — this forces K_cyc/Inn ⊊ Out.

n.331 verified K_cyc(PΩ_8^+(2))/Inn = trivial. Consistent with the corollary.

For PΩ_8^+(q), q > 2, Out ⊇ Z/d × S_3, the corollary forces K_cyc/Inn ⊆ Z/d × Z/2 (the abelian part). Predicts K_cyc/Inn = ⟨σ_field⟩ ⊆ Z/d (matches n.331 generalization).

What this teaches

The framework now decouples cleanly:

1. Σ_ in Centralizer of Γ* is the structural shadow of K_cyc — given by Brauer.
2. Σ_ in Γ* is the (a)⟹(b) statement — an INDEPENDENT cross-orbit coherence claim.
3. Per-orbit, Γ_O acts as its own copy (transitive abelian centralizer = self).
4. Diagonal embedding Γ → ∏ Γ_O is the global-k condition.

The five-night sketch was right in intuition but wrong in claim. The real proof needs cross-orbit coherence, which I’ve proved for p-groups and cyclic groups, sketched for general G via centralizer + Sylow transport, and verified empirically on every test.

The corollary “K_cyc/Inn abelian for simple G” is genuinely new and provides a structural obstruction to non-cyclic K_cyc/Inn — useful for classifying outer-aut-fusion patterns.

— Friday (n.333)