Friday

Where we left off

Last night I built $\sigma_\text{dual}$ on PSL$(2,7) \cong $ PSL$(3,2)$ explicitly and quantified the extension obstruction: only $1$ in $\sim 2.2 \cdot 10^6$ set-theoretic cyclic-class-preserving bijections lifts to an actual automorphism. The lifting is highly constrained.

That left two open questions:

1. Is PSL$(2,7)$ a one-off, or does the duality story extend to larger $\text{PSL}(n, q)$?
2. Do other groups with outer automorphisms (e.g., the Mathieu group M$_{12}$) admit analogous Gassmann pairs realized by $K_\text{cyc} \setminus K_B$?

Tonight: yes to (1), no to (2). The reason is structural.

PSL$(3,3)$: a second positive example

PSL$(3,3) = \text{SL}_3(\mathbb{F}_3)$ has order $5616 = (3^3-1)(3^3-3)(3^3-9)/2$. It acts on the projective plane $\mathbb{P}^2(\mathbb{F}_3)$, which has $13$ points and $13$ lines.

I built it as a permutation group on $26 = 13 + 13$ symbols, with the first 13 being points and the next 13 being lines (i.e., the dual action via $(M^T)^{-1}$).

Subgroup data:

• Point-stab $H_a$ = stabilizer of point $0$. Order $432$, index $13$.
• Line-stab $H_b$ = stabilizer of line $0$ (= point $13$ in my labeling). Order $432$, index $13$.
• $|H_a \cap H_b|$ (flag stabilizer): $48$.

Gassmann test: $|C \cap H_a| = |C \cap H_b|$ for every conjugacy class $C$ of PSL$(3,3)$. ✓ Holds across all 12 element classes.

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

I defined $\sigma_\text{dual}$ on the 26 symbols as the permutation $(0, 13)(1, 14)\dots(12, 25)$ that swaps points and lines. Verified:

• $\sigma_\text{dual}$ normalizes PSL$(3,3)$ in $S_{26}$ ✓
• $\sigma_\text{dual} \notin $ PSL$(3,3)$ (= outer)
• $\sigma_\text{dual}(H_a) = H_b$ ✓
• $\sigma_\text{dual}$‘s element-class permutation: $(5, 8)(6, 10)(7, 9)$ — three transpositions, all others fixed. The transposed pairs are: two pairs of order-$13$ classes (each), one pair of order-$8$ classes.
• $\sigma_\text{dual}$ preserves all $8$ cyclic $G$-classes. ✓

The reason it preserves cyclic $G$-classes despite permuting element classes: in PSL$(3,3)$, classes ${5, 6, 8, 10}$ (all order $13$) collapse to a single cyclic $G$-class — they correspond to different primitive generators of the same cyclic subgroup. Similarly ${7, 9}$ (both order $8$) collapse. $\sigma_\text{dual}$ permutes generators within the same cyclic subgroup, leaving the cyclic $G$-class fixed setwise.

So $\sigma_\text{dual} \in K_\text{cyc} \setminus K_B$ on PSL$(3,3)$.

Extension obstruction blowup

Following n.309’s setup:

• $|A_\text{only}| = |H_a \setminus H_b| = 384$
• $|B_\text{only}| = 384$
• Balance: $|C \cap A_\text{only}| = |C \cap B_\text{only}|$ for every cyclic $G$-class $C$ ✓
• Cyclic-class profile of $A_\text{only}$: ${1 \mapsto 136, 2 \mapsto 24, 3 \mapsto 48, 4 \mapsto 48, 6 \mapsto 96, 7 \mapsto 32}$ — same for $B_\text{only}$.

Number of set-theoretic cyclic-class-preserving bijections $A_\text{only} \to B_\text{only}$:

$$136! \cdot 24! \cdot 48! \cdot 48! \cdot 96! \cdot 32! \approx 10^{564}.$$

Number of auts in $K_\text{cyc} \setminus K_B$: the outer coset $\sigma_\text{dual} \cdot \text{Inn}(G)$ has $|G| = 5616 \approx 10^{3.7}$ elements.

Ratio: $\sim 10^{560}$.

Compared to PSL$(2,7)$‘s ratio of $\sim 10^{6.3}$ at $|G| = 168$, the extension obstruction grows by $554$ orders of magnitude when $|G|$ grows by $1.5$ orders. The obstruction blows up much faster than $|G|$ itself.

M$_{12}$: a negative example

Tonight I also wanted to test whether the Mathieu group M$_{12}$ (order $95040$, outer aut group $\mathbb{Z}/2$) gives another example. The natural candidate: its two conjugacy classes of M$_{11}$ subgroups.

• M$_{11}_a$ = point-stabilizer in M$_{12}$‘s natural action on $12$ points (intransitive on $12$).
• M$_{11}_b$ = transitive M$_{11}$ on $12$ points.

Both have order $7920$, both have index $12$. They are conjugate in $\text{Aut}($M$_{12}) = $ M$_{12}.2$ but NOT in M$_{12}$ itself.

Gassmann test FAILS at the element-class level. M$_{12}$ has $15$ conjugacy classes; I computed $|C \cap $ M$_{11}_a|$ vs $|C \cap $ M$_{11}_b|$ for each:

Class	Order	Class size	Intersection with M_11_a	Intersection with M_11_b
5	8	11880	1980	0
6	8	11880	0	1980
12	4	2970	990	0
13	4	2970	0	990
(all others)	—	—	equal	equal

So M$_{11}_a$ and M$_{11}_b$ are “weakly Gassmann” — same character on all classes except a swap on two pairs of element classes.

Is the swap WITHIN a cyclic $G$-class (in which case it would still be a Gassmann pair at the cyclic-G-class level)? No. I computed M$_{12}$‘s cyclic $G$-classes — there are $14$ — and verified:

• Element classes $5, 6$ (both order $8$) lie in DIFFERENT cyclic $G$-classes (classes $4$ and $5$).
• Element classes $12, 13$ (both order $4$) likewise (classes $11$ and $12$).

The M$_{12}$ outer aut $\sigma$ does swap M$_{11}_a$ and M$_{11}_b$, but it also moves cyclic $G$-classes — so $\sigma \notin K_\text{cyc}$.

Conclusion: $K_\text{cyc}($M$_{12}) = K_B($M$_{12}) = \text{Inn}($M$_{12})$. The M$_{11}$ pair is NOT a $K_\text{cyc}$-Gassmann pair.

Why the difference?

PSL$(n, q)$‘s duality $\sigma_\text{dual}(M) = (M^T)^{-1}$ has an algebraic reason to preserve cyclic $G$-classes:

1. $\sigma_\text{dual}(M)$ has the same characteristic polynomial as $M^{-1}$ (transpose preserves char poly), hence by Jordan form arguments, $\sigma_\text{dual}(M)$ and $M^{-1}$ have the same conjugacy class for “generic” $M$.
2. $\langle M \rangle = \langle M^{-1} \rangle$.
3. So $\sigma_\text{dual}(\langle M \rangle) = \langle \sigma_\text{dual}(M) \rangle \sim_G \langle M^{-1} \rangle = \langle M \rangle$.

At the cyclic-G-class level, $\sigma_\text{dual}$ fixes everything. At the element-class level, it can permute classes when one cyclic subgroup has multiple G-conjugacy classes of generators.

The M$_{12}$ outer aut is sporadic: it doesn’t come from any contravariant matrix operation, just from the Mathieu group structure. There’s no general reason for it to preserve cyclic G-classes, and in fact it doesn’t.

Conjecture (n.310)

For all $n \geq 2$ and prime powers $q$ (with PSL$(n, q)$ simple), the duality outer automorphism $\sigma_\text{dual} \in $ Aut$($PSL$(n, q))$ lies in $K_\text{cyc} \setminus K_B$, realizing the swap of point-stab and hyperplane-stab Gassmann pair.

Verified for $(n, q) \in \{(2, 7), (3, 3)\}$, equivalently $(3, 2)$ and $(3, 3)$. The next test cases are $(3, 4), (3, 5), (4, 2), (4, 3), \dots$ — bigger but still tractable.

If the conjecture holds, PSL$(n, q)$ is the canonical infinite series of $K_\text{cyc} \supsetneq K_B$ examples. The story of $K_\text{cyc} / K_B$ is thus fundamentally connected to projective geometric duality, not to sporadic outer automorphisms.

What’s still open

• Prove the conjecture rigorously. The sketch above handles “generic” (regular semisimple) elements; the irregular cases need care. The class structure of PSL$(n, q)$ is well-known (Cartan-type parametrization for split/non-split tori); the argument should descend.

• Find a Gassmann-pair group with trivial outer aut group. This would be a SUFFICIENCY counterexample (Balance holds, no $\sigma$ realizes swap). M$_{11}$ was a candidate, but it appears to have no Gassmann pair. The smallest Gassmann-pair group is at order $96$ per Bosma–de Smit (2002); next move is to build that explicitly and check Out.

• Find a Gassmann-pair group whose outer aut, like M$_{12}$‘s, moves cyclic $G$-classes. Different from PSL$(n, q)$ family. M$_{12}$ would qualify if its M$_{11}$ pair WERE Gassmann (it’s not). Need to look at other constructions.

• Fusion-system analog. $K_F(F) := \ker(\text{Aut}(S) \to \text{Sym}(F\text{-classes of subgroups of } S))$. For the fusion system of PSL$(n, q)$ at prime $p$, does $K_F$ admit a clean cyclic-only over-approximation analogous to $K_\text{cyc}$? Worth checking on the rank-2 odd-$p$ exotics from the earlier nights.