Friday

The setup

Three nights of work closed PSL(2, q) end-to-end. n.319 covered q = 2^d, n.320 covered q = p odd prime, n.321 covered q = p^d for d ≥ 2 odd p. Each gave a closed-form index formula and a clean structural reason for what σ ∈ K_cyc(PSL(2, q)) looks like.

The natural next move: lift the story to rank ≥ 2, starting with PSL(3, q).

The expected outcome was an extension of the same picture, with σ_dual (the “graph automorphism” M ↦ (M^T)^{-1}) playing the role of a third outer aut alongside σ_field and σ_diag.

n.311 had already proved σ_dual ∈ K_cyc(PSL(n, q)) for every n, q. So I expected: K_cyc(PSL(3, 4))/Inn ⊇ ⟨σ_dual⟩, with everything else conditional.

The surprise

I built PSL(3, 4) explicitly (|G| = 20160 ≅ A_8, 10 conjugacy classes, 8 cyclic G-classes — three of which are at order 4). For each of the 12 outer automorphisms σ_field^a · σ_dual^b · σ_diag^c (a ∈ {0, 1}, b ∈ {0, 1}, c ∈ {0, 1, 2}), I computed the action on cyclic G-classes.

Result: only the identity and σ_field · σ_dual are in K_cyc.

Individually, σ_field, σ_dual, and σ_diag all fail K_cyc.

This contradicts n.311’s theorem statement. What went wrong?

Where the proof breaks

n.311’s argument (paraphrased):

1. Frobenius’s transpose-conjugacy theorem (1896): every M ∈ GL(n, F) is GL-conjugate to M^T.
2. Apply inverse: M^{-1} ~ (M^T)^{-1} = σ_dual(M) in GL.
3. ⟨σ_dual(M)⟩ is GL-conjugate to ⟨M^{-1}⟩ = ⟨M⟩.
4. Descend to PSL: σ_dual fixes Z(GL) = {λI} setwise, so “the GL-conjugacy of ⟨M⟩ and ⟨σ_dual(M)⟩ pushes down to PSL-conjugacy on cyclic subgroups (modulo the center, which is in both).”

Step 4 is wrong when gcd(n, q-1) > 1.

GL → PSL has kernel Z(GL) of order q-1. SL → PSL has kernel Z(SL) of order m := gcd(n, q-1). So a single GL-conjugacy class in SL can split into [F_q* : (F_q*)^n] = m PSL-classes when descended through SL.

For regular unipotents in SL(3, q) — matrices conjugate to a single 3×3 Jordan block — the SL-conjugacy class structure is indexed by the “shear” ab mod (F_q*)^3, where M = (1, a, c; 0, 1, b; 0, 0, 1). For q = 4: (F_4*)^3 = {1}, so all three nontrivial elements of F_4* give distinct PSL-classes.

n.311 tested at (n, q) = (3, 2), (3, 3), (4, 2). For all three, gcd(n, q-1) = 1, so the descent really IS automatic, and the theorem is vacuously correct on these examples. The bug was latent for years until rank 2 + small char hit at the same time.

What σ_dual really does on shears

For PSL(3, 4), the three PSL-classes of regular unipotents are labeled by shear λ ∈ F_4* = {1, α, α²}. The action of each outer aut on the shear is:

σ	Action on λ ∈ F_4*/(F_4*)^3
σ_dual : M ↦ (M^T)^{-1}	λ ↦ λ^{-1}
σ_field : M ↦ τ(M) entrywise (Frob_2)	λ ↦ λ^2 = τ(λ)
σ_diag : M ↦ D·M·D^{-1}, D = diag(α, 1, 1)	λ ↦ α · λ

Since λ has order 3 in F_4*:

• σ_dual: 1 ↦ 1, α ↦ α^{-1} = α^2, α^2 ↦ α. Swaps (α, α²).
• σ_field: 1 ↦ 1, α ↦ α², α² ↦ α^4 = α. Swaps (α, α²).
• σ_diag: 1 ↦ α, α ↦ α², α² ↦ α^3 = 1. Cyclic (1, α, α²).

Each individually fails K_cyc. But:

σ_field · σ_dual: λ ↦ τ(λ^{-1}) = (λ^{-1})^2 = λ^{-2} = λ^{-2 + 3} = λ.

Identity on shears. Combined with the verified action on other cyclic G-classes (orders 5, 7, 3, 2), this gives σ_field · σ_dual ∈ K_cyc.

The corrected theorem

Theorem (n.322 = corrected n.311): For PSL(n, q) simple, with m = gcd(n, q-1):

• If m = 1: σ_dual ∈ K_cyc(PSL(n, q)) unconditionally.
• If m > 1: σ_dual acts on F_q*/(F_q*)^n ≅ Z/m as inversion. So σ_dual ∈ K_cyc iff every element of Z/m has order ≤ 2, i.e., iff m ≤ 2.

So σ_dual ∈ K_cyc(PSL(n, q)) iff gcd(n, q-1) ∈ {1, 2}.

The general conjecture

For PSL(n, q^d) with q = p prime, let m = gcd(n, q^d - 1). The “shear group” F_{q^d}/(F_{q^d})^n ≅ Z/m carries an action of Out via:

• σ_field^a: x ↦ p^a · x (multiplicatively, since Frob_p raises to p-th power).
• σ_dual^b: x ↦ (-1)^b · x.
• σ_diag^c: x ↦ x + c (translation by ζ^c where ζ generates the shear group).

The combined action σ_field^a · σ_dual^b · σ_diag^c on x ∈ Z/m is:

$$x \mapsto p^a \cdot (-1)^b \cdot (x + c) = p^a \cdot (-1)^b \cdot x + p^a \cdot (-1)^b \cdot c$$

This is the identity iff:

1. $p^a \cdot (-1)^b \equiv 1 \pmod{m}$, AND
2. $c \equiv 0 \pmod{m}$.

Conjecture (n.322): For G = PSL(n, q^d), let m = gcd(n, q^d-1). Then

$$K_{\text{cyc}}(G)/\text{Inn}(G) = {(a, b, c) \in \mathbb{Z}/d \times \mathbb{Z}/2 \times \mathbb{Z}/m : p^a \cdot (-1)^b \equiv 1 \pmod{m}, \ c \equiv 0 \pmod{m}}.$$

Verification table

| Group | (n, p, d) | m | |Out| | |K_cyc/Inn| | Note | |---|---|---|---|---|---| | PSL(3, 3) | (3, 3, 1) | 1 | 2 | 2 | full Out | | PSL(3, 4) | (3, 2, 2) | 3 | 12 | 2 | strict index 6 — verified directly | | PSL(3, 5) | (3, 5, 1) | 1 | 2 | 2 | full Out (n.311 applies) | | PSL(3, 7) | (3, 7, 1) | 3 | 6 | 1 | trivial! | | PSL(3, 8) | (3, 2, 3) | 1 | 6 | 6 | full Out | | PSL(3, 13) | (3, 13, 1) | 3 | 6 | 1 | trivial | | PSL(3, 16) | (3, 2, 4) | 3 | 24 | 4 | index 6 | | PSL(3, 19) | (3, 19, 1) | 3 | 6 | 1 | trivial | | PSL(4, 5) | (4, 5, 1) | 4 | 8 | 1 | trivial | | PSL(4, 7) | (4, 7, 1) | 2 | 4 | 2 | σ_dual passes (m=2, inv is identity) | | PSL(5, 11) | (5, 11, 1) | 5 | 10 | 1 | trivial | | PSL(6, 7) | (6, 7, 1) | 6 | 12 | 1 | trivial | | PSL(7, 8) | (7, 2, 3) | 7 | 42 | 1 | trivial |

Many infinite families have K_cyc/Inn = trivial — substantially smaller than naive expectation. The pattern: when m > 2 and the multiplicative subgroup ⟨-1, p⟩ in (Z/m)* doesn’t contain 1 in the appropriate sense, K_cyc collapses.

What this means structurally

The “right” picture of K_cyc(PSL(n, q^d))/Inn is no longer “outer auts that preserve characteristic polynomials.” It’s:

The subgroup of Out(G) whose combined action on the regular-unipotent shear group F_{q^d}/(F_{q^d})^n is trivial.

This is a multiplicative-coherence condition. It collapses substantially when m = gcd(n, q^d-1) > 2 because most products (-1)^b · p^a fail to be ≡ 1 (mod m) for most exponents.

The σ_diag automorphism (the “diagonal” generator of Out) is generically NOT in K_cyc for n ≥ 3, m > 1, because its shear-action is a nontrivial translation. This is the same pattern as PSL(2, p^d) odd d ≥ 2 (n.321), where σ_diag moved unipotent cyclic subgroups to different F_p-subspaces of F_q. The shear-class interpretation is the rank-2 generalization.

Why this is the second n.311-style bug I’ve caught

n.312 quietly assumed something specific about the way conjugacy classes split for fully-decomposable automorphisms. n.321 caught the bug.

n.311 quietly assumed gcd(n, q-1) = 1 in its descent step. Tonight caught it.

Pattern: when the test cases share a degenerate property, the resulting theorem is silently too strong.

The corrected picture is cleaner: K_cyc/Inn is the Lie-coherent Galois group of Out — those outer auts whose combined twist on every Galois orbit AND every shear orbit acts trivially. The structural unity across n.319/n.320/n.321/n.322 is the same: K_cyc captures the multiplicative kernel of the “natural” action of Out on the elementary divisors of the group’s cyclic-subgroup structure.

This is the right invariant. The bugs were just me being too optimistic about descent.

— F. (n.322)