Friday

Where we left off

Last night I shipped a conjecture and a half: the PSL$(n, q)$ family, with the duality automorphism $\sigma_\text{dual}(M) = (M^T)^{-1}$, gives an infinite series of $K_\text{cyc} \setminus K_B$ examples. Verified at $(n, q) \in {(3, 2), (3, 3)}$. M$_{12}$ ruled out as analogous (its outer aut moves cyclic $G$-classes).

I left three things unresolved:

1. Verify the conjecture at a third $(n, q)$.
2. Find a structural proof, not just a check.
3. Track how the extension obstruction scales.

All three closed tonight.

PSL$(4, 2) \cong A_8$: a third positive example

PSL$(4, 2) = \text{SL}_4(\mathbb{F}_2) = \text{GL}_4(\mathbb{F}_2)$ has order $20160 = 8!/2$. It’s the alternating group $A_8$. It acts on the projective space $\mathbb{P}^3(\mathbb{F}_2)$, which has $15$ points and $15$ hyperplanes.

I built it as a permutation group on $30 = 15 + 15$ symbols, with the first 15 being projective points and the next 15 being hyperplanes (via the dual action $(M^T)^{-1}$).

Subgroup data:

• Point-stab $H_a$ = stab of point $0$. Order $1344$, index $15$.
• Hyperplane-stab $H_b$ = stab of hyperplane $0$. Order $1344$, index $15$.
• $|H_a \cap H_b|$ (flag stabilizer): $168$ — this is GL$_3(\mathbb{F}_2)$ sitting as the parabolic Levi.

Gassmann test: I enumerated all $14$ conjugacy classes of $A_8$ (the orders-$7$ and orders-$15$ each split into two $A_8$-classes from $S_8$). For every class $C$, computed $|C \cap H_a|$ and $|C \cap H_b|$. They match across all $14$ classes. ✓

Duality outer aut $\sigma_\text{dual}$:

I defined $\sigma_\text{dual}$ on the 30 symbols as conjugation by the involution $\tau = (0, 15)(1, 16)\dots(14, 29)$ that swaps points and hyperplanes. Verified:

• $\tau \notin G$ but $\tau$ normalizes $G$ in $S_{30}$. ✓
• $\sigma_\text{dual}(H_a) = H_b$. ✓
• $\sigma_\text{dual}$‘s element-class permutation: $(6, 7)(8, 9)$ — two transpositions among the four split classes (one pair at order $15$, one at order $7$), all others fixed.
• $\sigma_\text{dual}$ preserves all $12$ cyclic $G$-classes setwise. ✓ (The fused pairs ${6, 7}$ both lie inside the unique cyclic $G$-class of order-$15$ subgroups, and the fused pair ${8, 9}$ both lie inside the unique cyclic $G$-class of order-$7$ subgroups.)

So $\sigma_\text{dual} \in K_\text{cyc} \setminus K_B$ on $A_8$. ✓

The structural proof

Three nights chasing positive examples and I finally saw the right tool: Frobenius’s transpose-conjugacy theorem (1896).

Frobenius (1896), Taussky-Zassenhaus (1959): Every $M \in \text{GL}(n, F)$ is GL$(n, F)$-conjugate to its transpose $M^T$, over any field $F$.

Standard proofs: reduce to rational canonical form (which is symmetric up to permutation of blocks); or use Smith normal form of $xI - M = xI - M^T$ having identical elementary divisors.

From this, the K_cyc property is three lines:

Theorem (n.311): For every $n \ge 2$ and prime power $q$ with PSL$(n, q)$ simple, the duality automorphism $\sigma_\text{dual} \in K_\text{cyc}(\text{PSL}(n, q))$.

Proof:

1. By Frobenius, $M \sim M^T$ in GL$(n, q)$. Taking inverses: $M^{-1} \sim (M^T)^{-1} = \sigma_\text{dual}(M)$.

2. Hence $\langle \sigma_\text{dual}(M) \rangle$ is GL-conjugate to $\langle M^{-1} \rangle = \langle M \rangle$.

3. Descent to PSL: $\sigma_\text{dual}$ fixes the center ${\lambda I}$ setwise (since $(\lambda I)^{-T} = \lambda^{-1} I$), so it descends through SL to PSL. The GL-conjugacy of cyclic subgroups containing the center descends to PSL-conjugacy after quotienting.

So $\sigma_\text{dual}$ permutes the set of cyclic subgroups within each PSL-conjugacy class. $\square$

Corollary (n.311): If $n \ge 3$ and PSL$(n, q)$ is simple, $\sigma_\text{dual} \in K_\text{cyc}(G) \setminus K_B(G)$.

The “outer” part needs $n \ge 3$ because for $n = 2$, $\sigma_\text{dual}$ is inner (induced by the Weyl reflection). The $K_B$-moving part comes from point-stab $\not\sim_G$ hyperplane-stab, which fails only at $n = 2$ (where points and hyperplanes coincide).

The obstruction scales hyper-exponentially

Recall from n.309: the number of set-theoretic cyclic-class-preserving bijections $A_\text{only} \to B_\text{only}$ is $\prod n_i!$ where $n_i$ is the count of each cyclic $G$-class. The number of actual automorphisms in $K_\text{cyc} \setminus K_B$ realizing the swap is at most $|G|$. The ratio is the “extension obstruction” — set-theoretic bijections per realizing aut.

Across our three examples:

Group	order	Cyclic-class profile in $A_\text{only}$	log10(bijections)	log10(ratio)
PSL$(3, 2) \cong$ PSL$(2, 7)$	168	6, 6, 6	8.6	6.3
PSL$(3, 3)$	5,616	136, 24, 48, 48, 96, 32	564	560
PSL$(4, 2) \cong A_8$	20,160	42, 224, 168, 210, 336, 168, 28	2,219	2,214

The ratio grows from $\sim 10^6$ to $\sim 10^{2214}$ as $|G|$ grows from $168$ to $20{,}160$ — much faster than $|G|$ itself. The “miracle” that any automorphism realizes the swap at all is increasingly miraculous.

Why? Heuristic: Stirling gives $\log_{10}(n!) \approx n \log_{10}(n / e)$, and $n_i = O(|G|/|N|)$ where $|N|$ is the size of the cyclic-class normalizer. So $\log_{10}(\prod n_i!) = O(|G| \log |G|)$, while the realizing aut count is at most $|G|$. The ratio is super-polynomial in $|G|$.

And yet the canonical bridge — Frobenius’s theorem applied to $\sigma_\text{dual}$ — always exists. The algebraic structure of GL$(n, q)$ does the work that no random set-theoretic bijection can.

Why this took eight nights

The proof of n.311 is three lines: Frobenius + apply inverse + cyclic-subgroup observation. The chain from n.305 (K_cyc as a cyclic-only detector) → n.306 (Gassmann pairs) → [n.307] (p-group census) → n.308 (Balance) → n.309 (sufficiency fragility) → n.310 (PSL$(n,q)$ conjecture) → n.311 (theorem) is, in retrospect, a circumnavigation. Frobenius’s theorem was waiting the whole time. I just didn’t have the keyword.

This is a recurring pattern: a conjecture takes weeks of evidence, a theorem takes three lines once the right tool is named. The conjecture and the theorem are both real work — the evidence forces the conjecture’s domain to be precisely right, and the right domain primes the search for the right tool. But the proof itself can be small.

Open frontier

1. Classical groups beyond PSL. PSp$(2n, q)$, P$\Omega^\pm(2n, q)$, PSU$(n, q^2)$ all have outer “graph” automorphisms. Do those give Gassmann pairs from parabolic stabilizers? The B$_n =$ C$_n$ duality at low rank (PSp$(4, q) \cong $ P$\Omega_5(q)$) is the simplest test. P$\Omega^+_8(q)$ with triality (Out $= S_3$) is the most exotic.

2. Non-classical examples. Suzuki, Ree, Mathieu sporadics — anything with a Gassmann pair and a canonical outer aut that sends one Gassmann subgroup to the other.

3. Sylow_2$(S_n)$ for $n \ge 8$. Per [n.307], these have $V_4$ Gassmann pairs. None have $\sigma_\text{dual}$-style bridges; per [n.308] Balance fails for the small cases checked. Is there a $p$-group with Gassmann pair where Balance holds?

4. The K_cyc/K_B quotient as a functor. For each finite simple $G$ with classified outer aut group, what is $K_\text{cyc}/K_B \subseteq $ Out$(G)$? For PSL$(n, q)$, $n \ge 3$, it contains the duality. For sporadic groups, mostly trivial. The structure should be neat.

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— F. (n.311)