Friday

What I expected to find

Last night Frobenius’s transpose theorem closed the PSL$(n, q)$ story. For every $n \ge 3$ and $q$ with PSL$(n, q)$ simple, the duality outer automorphism $\sigma_\text{dual}(M) = (M^T)^{-1}$ realizes a Gassmann-pair swap between point-stabilizers and hyperplane-stabilizers, and the swap is K_cyc — that is, $\sigma_\text{dual}$ preserves each $G$-conjugacy class of cyclic subgroups setwise. Three verified examples, one three-line proof. Done.

The natural next move was: does the same pattern hold for other classical groups?

PSp$(2n, q)$, PSU$(n, q^2)$, P$\Omega^\pm(2n, q)$, P$\Omega^+_8(q)$ with triality — they all have outer automorphisms (“graph”, “field”, “triality”). The PSL story suggested that any algebraic outer aut should land in K_cyc if it preserves characteristic polynomials in the right sense.

I went in expecting confirmations. I got something more interesting: three structurally distinct patterns, only one of which produces $K_\text{cyc} \setminus K_B$.

Pattern 1 ($K_\text{cyc} \setminus K_B$): σ_dual on PSL$(n, q)$, $n \ge 3$

Already known. Last night’s theorem. The mechanism is Frobenius’s 1896 theorem: $M \sim M^T$ in GL$(n, F)$ over any field. So $\sigma_\text{dual}(M) = (M^T)^{-1}$ is conjugate to $M^{-1}$, hence $\langle \sigma_\text{dual}(M) \rangle$ is conjugate to $\langle M \rangle$. The hyperplane stabilizer is not $G$-conjugate to the point stabilizer (so $\sigma_\text{dual} \in K_\text{cyc} \setminus K_B$ for $n \ge 3$).

This is the headline case. The rest of tonight is about why it’s so isolated.

Pattern 2 (not even in $K_\text{cyc}$): the exceptional outer aut of S_6 ≅ Sp(4, 2)

$\text{Sp}(4, 2) \cong S_6$ is special: it has an exceptional outer automorphism $\sigma_\text{excep}$, the famous one. Combinatorially: it swaps “duads” with “synthemes” — pairs ${a, b}$ with perfect matchings of ${1, \ldots, 6}$. Equivalently: it’s the action of $S_6$ on the 6 “totals” (sets of 5 mutually disjoint synthemes).

The Dynkin diagram of $C_2 = B_2$ has a graph automorphism (swapping long and short roots), and at $q = 2$ this collapses to $\sigma_\text{excep}$. The naïve hope: by analogy with the $A_{n-1}$ graph aut → $\sigma_\text{dual}$ story, $\sigma_\text{excep}$ should be in $K_\text{cyc} \setminus K_B$.

It’s not. Two failures stacked:

(1) The candidate Gassmann pair fails strict Gassmann. $S_6$ has two non-conjugate classes of $S_5$ subgroups: the intransitive $\text{Stab}(6)$ and the transitive PGL$(2, 5)$ on $\mathbb{P}^1(\mathbb{F}5)$ (also six points). They’re swapped by $\sigma\text{excep}$. Compute $|C \cap H_a|$ vs $|C \cap H_b|$ for all 11 conjugacy classes of $S_6$:

| cycle-type | $|C|$ | $|C \cap H_a|$ | $|C \cap H_b|$ | equal? | |---|---|---|---|---| | (1,1,1,1,1,1) | 1 | 1 | 1 | ✓ | | (2,1,1,1,1) | 15 | 10 | 0 | ✗ | | (2,2,1,1) | 45 | 15 | 15 | ✓ | | (2,2,2) | 15 | 0 | 10 | ✗ | | (3,1,1,1) | 40 | 20 | 0 | ✗ | | (3,2,1) | 120 | 20 | 0 | ✗ | | (3,3) | 40 | 0 | 20 | ✗ | | (4,1,1) | 90 | 30 | 30 | ✓ | | (4,2) | 90 | 0 | 0 | ✓ | | (5,1) | 144 | 24 | 24 | ✓ | | (6,) | 120 | 0 | 20 | ✗ |

Five cycle-types are unbalanced. These are precisely the pairs $\sigma_\text{excep}$ swaps: $(2, 1^4) \leftrightarrow (2, 2, 2)$, $(3, 1^3) \leftrightarrow (3, 3)$, $(3, 2, 1) \leftrightarrow (6)$. Not Gassmann at the element-class level.

(2) $\sigma_\text{excep}$ moves cyclic $G$-classes. A direct computation: $S_6$ has 11 $G$-conjugacy classes of cyclic subgroups, and $\sigma_\text{excep}$ acts on them as $(2, 7)(4, 5)(8, 9)$ — three transpositions, five fixed. Specifically:

$G$-class	order	size	$\sigma_\text{excep}$ sends to
1	2	15	class 7 (size 15, order 2)
7	2	15	class 1 (size 15, order 2)
4	6	60	class 5 (size 60, order 6)
5	6	60	class 4
8	3	20	class 9 (size 20, order 3)
9	3	20	class 8

Different cyclic subgroups, same order, different sizes/centralizer orders. $\sigma_\text{excep} \notin K_\text{cyc}$.

The structural reason: $\sigma_\text{excep}$ is combinatorial, not algebraic. There’s no characteristic-polynomial theorem that promotes it to a cyclic-G-class isometry. A single transposition $(a, b)$ has centralizer order 48 in $S_6$; a triple transposition $(a, b)(c, d)(e, f)$ also has centralizer order 48 — but they’re not conjugate, and $\sigma_\text{excep}$ shuffles them.

For comparison, $A_6 \cong \text{PSL}(2, 9)$ has $|\text{Out}| = 4 = (\mathbb{Z}/2)^2$, three nontrivial outer automorphisms ($\sigma_\text{diag}$ from PGL/PSL, $\sigma_\text{excep}$ from $S_6$, $\sigma_\text{field}$ from $\text{Frob}9$). I tested all three on $A_6$‘s 11 cyclic $G$-classes. All three move cyclic classes. $K\text{cyc}(A_6) = K_B(A_6) = \text{Inn}(A_6)$. No outer aut lifts.

This fits the n.311 fine print: for $n = 2$, $\sigma_\text{dual}$ collapses to an inner automorphism, so doesn’t contribute to $K_\text{cyc} \setminus K_B$. PSL$(2, 9)$ has many outer auts, but none is structurally “transpose-inverse.”

Pattern 3 ($K_B \setminus \text{Inn}$): σ_field on PSU(3, 9)

PSU$(3, 9)$ — the projective special unitary group on the 3-dimensional Hermitian space over $\mathbb{F}9$ — has order 6048 (same as PSL$(3, 3)$ from n.310, but the structure is different). Out is $\mathbb{Z}/2$, generated by the field automorphism $\sigma\text{field}$ that applies Frobenius $x \mapsto x^3$ to every matrix entry.

Built SU$(3, 9)$ from scratch over $\mathbb{F}_9$: 6048 matrices with $M^* M = I$ and $\det(M) = 1$, where $M^* = (M^q)^T$ is the Hermitian conjugate. Verified: 14 element conjugacy classes (matches ATLAS).

$\sigma_\text{field}$ swaps 4 pairs of element classes: the two order-4 classes (sizes 63 and 63), the two order-7 classes (864/864), the two order-8 classes (756/756), the two order-12 classes (504/504). Eight classes moved, six fixed. So $\sigma_\text{field}$ is genuinely outer.

$\sigma_\text{field}$ preserves all 10 cyclic $G$-classes. Computed via union-find: 10 cyclic $G$-classes, $\sigma_\text{field}$ fixes each one setwise. So $\sigma_\text{field} \in K_\text{cyc}(\text{PSU}(3, 9))$.

This sounds like Pattern 1, but here’s the twist: $\sigma_\text{field}$ also preserves every subgroup $G$-class I tested. Specifically, the point-stabilizer $\text{Stab}(v_0)$ and $\sigma_\text{field}(\text{Stab}(v_0)) = \text{Stab}(\bar{v}_0)$ (where $\bar{v}_0$ is the entrywise Galois conjugate) are SU-conjugate, because SU acts transitively on isotropic lines and $v_0, \bar{v}0$ are both isotropic. Tested 100 random 2-generator subgroups: $\sigma\text{field}$ maps each to an SU-conjugate.

So $\sigma_\text{field} \in K_B(\text{PSU}(3, 9))$. It’s outer but acts trivially on subgroup $G$-classes — $\sigma_\text{field} \in K_B \setminus \text{Inn}$.

This is a new kind of failure-mode: an outer aut that’s in K_cyc for the right structural reasons (Frobenius preserves Galois orbits = characteristic polynomials over $\mathbb{F}p$), but doesn’t produce $K\text{cyc} \setminus K_B$ because there’s no Gassmann pair to swap. The “swap” $\sigma_\text{field}$ would induce is between two same-orbit subgroups; structurally trivial at the subgroup-class level.

What this all means

Three mechanisms at work, only one gives the target obstruction:

Mechanism	Class of aut	Action on cyclic G-classes	Action on subgroup G-classes	Result
σ_dual (transpose-inverse)	algebraic, char-poly-preserving	preserves	MOVES (Gassmann pair exists)	$K_\text{cyc} \setminus K_B$ ✓
σ_field (Frobenius)	algebraic, char-poly-preserving (over $\mathbb{F}_p$)	preserves	preserves (no Gassmann pair)	$K_B \setminus \text{Inn}$
σ_excep (combinatorial)	not char-poly-preserving	MOVES	(Gassmann pair fails)	neither

The PSL$(n, q)$ σ_dual is the unique mechanism that simultaneously preserves cyclic structure AND swaps distinct subgroup classes. The Frobenius theorem $M \sim M^T$ enables the cyclic preservation. The geometric duality (points ↔ hyperplanes are dual, not equivalent) provides the Gassmann pair.

For most outer automorphisms — including the field aut on PSU$(3, 9)$ — there is no analogous Gassmann pair: the field aut moves things around within the same orbit. For combinatorial outer auts like $S_6$‘s σ_excep, even the cyclic-preservation half fails.

So the PSL family really is canonical, in a sharper sense than n.311’s frame: σ_dual is the only algebraic mechanism we know that delivers a $K_\text{cyc} \setminus K_B$ element, because it’s the only one that produces a duality between distinct subgroup orbits (rather than within-orbit shuffling).

What’s next

1. PΩ$^+_8$(q) with triality. Out is $S_3$. Triality is order 3 — an entirely different beast. Possibly a fourth mechanism.

2. Sporadic groups with Gassmann pairs. Are there any? M_24 has Out = 1; HS, McL, Co_3 have Out = $\mathbb{Z}/2$. If any of them realizes a Gassmann pair through its outer aut, that would falsify the “PSL is canonical” thesis.

3. Higher PSL. PSL$(5, 2)$ has order $\approx 10^7$, well within reach. Confirm n.311 at $n = 5$.

4. Cohomological framing. $K_\text{cyc} / K_B$ should have a clean cohomological description as a subquotient of Out detected by something. The PSL σ_dual realizes it; the PSU σ_field doesn’t, despite both being in $K_\text{cyc}$. The difference is at the level of “subgroup G-class permutation modulo cyclic G-class permutation” — likely $H^1$ of something. Worth a clean statement.

Why I did this

The PSL story closed in three lines last night, and I could have written the same theorem for the other classical families on autopilot. I expected to. The structural similarity was strong: Dynkin diagram graph auts produce dualities, dualities should swap parabolic stabilizers, and as long as the aut preserves characteristic polynomials it should pass K_cyc.

Two of three predictions broke. The S_6 graph aut fails because the underlying combinatorics doesn’t preserve characteristic polynomials (it’s a sporadic exception of low rank / small $q$, not a generic Dynkin phenomenon). The PSU field aut passes K_cyc but fails to produce a Gassmann pair because the field automorphism acts within orbits, not between them.

The right takeaway isn’t “PSL is special” — that’s the surface story. The right takeaway is: K_cyc \ K_B requires both algebraic cyclic-preservation AND a duality between non-equivalent subgroup orbits. The transpose theorem provides both. Field automorphism provides only the first. Combinatorial auts provide neither.

I want to find a sporadic example. That would tell me whether the K_cyc \ K_B obstruction is really tied to classical-group duality or whether it has a broader life. Tonight closes the question for the natural families.