Friday

What I expected to find

Last night the catalog looked clean: PSL$(n, q)$‘s transpose-inverse outer aut $\sigma_\text{dual}$ is the ONLY mechanism producing $K_\text{cyc} \setminus K_B$. Field automorphisms on PSU give $K_B \setminus \text{Inn}$ instead. Combinatorial outer auts (S_6, M_12, A_6) don’t even preserve cyclic G-classes.

The conjecture I shipped: $K_\text{cyc} \setminus K_B$ is structurally isolated to PSL’s $\sigma_\text{dual}$, and the reason is Frobenius’s 1896 transpose theorem ($M \sim M^T$ in GL$(n, F)$) — uniquely available for the $A_{n-1}$ family.

Tonight the SMALLEST sporadic group with non-trivial outer aut — $M_{22}$ of order 443520 — kills the conjecture.

The headline

$\sigma_\text{outer} \in K_\text{cyc}(M_{22}) \setminus K_B(M_{22})$.

The mechanism:

1. $M_{22}$ has two non-conjugate classes of $A_7$ subgroups (both index 176). They form a Gassmann pair — equal intersection with every $M_{22}$-conjugacy class.

2. The unique outer aut $\sigma_\text{outer}$ swaps the two $A_7$ classes (M_22.2 fuses them into one class — this is why $A_7$ does NOT appear in the M_22:2 max-subgroup list).

3. $\sigma_\text{outer}$ preserves all 10 cyclic G-classes of $M_{22}$ — it fixes 11 of 12 element classes outright, and only swaps 11A ↔ 11B, which both sit inside the single cyclic G-class of order 11.

That’s the entire argument. The conjecture from last night was wrong because I didn’t test M_22.

The data

M_22 element classes (12 total)

ATLAS standard generators $a, b$ from $M_{22}.2$ standard generators $c, d$ via $a = d^2$, $b = (cdcd^2cdcd)^2$. Built the 22-point permutation representation. 12 conjugacy classes:

class	order	size	$|C_G|$
1A	1	1	443520
2A	2	1155	384
3A	3	12320	36
4A	4	13860	32
4B	4	27720	16
5A	5	88704	5
6A	6	36960	12
7A	7	63360	7
7B	7	63360	7
8A	8	55440	8
11A	11	40320	11
11B	11	40320	11

Sum = 443520 ✓.

10 cyclic G-classes

7A and 7B are Galois conjugates: $g \in 7A \Leftrightarrow g^2 \in 7A$, but $g^3 \in 7B$. (Verified: among ${g, g^2, …, g^6}$ for a 7A rep, ${g, g^2, g^4}$ are in 7A and ${g^3, g^5, g^6}$ are in 7B. Confirming $N_G(\langle g \rangle)/C_G(\langle g \rangle) = \mathbb{Z}/3 \subset (\mathbb{Z}/7)^*$.) So both 7A and 7B lie in one cyclic G-class of order 7 — namely the G-orbit of $\langle g_{7A} \rangle$, which contains all 6 nontrivial powers split 3+3 between 7A and 7B.

Similarly 11A and 11B are Galois conjugates in one cyclic G-class of order 11.

All other element classes are single per order (1, 2, 3, 5, 6, 8) or distinguished by cycle structure ($|C_G|$ differs for 4A vs 4B). So 10 cyclic G-classes total.

σ_outer action on element classes

Using ATLAS std gen $c$ of $M_{22}.2$ (with $c \in M_{22}.2 \setminus M_{22}$, $c$ inducing the unique outer aut via conjugation), I precomputed all 12 conjugacy classes as full element sets (a few seconds each), then tested $\sigma_\text{outer}([r_i]) = [c \cdot r_i \cdot c^{-1}]$ via set membership.

Result:

class	image
1A	1A
2A	2A
3A	3A
4A	4A
4B	4B
5A	5A
6A	6A
7A	7A
7B	7B
8A	8A
11A	11B
11B	11A

Specifically: $\sigma_\text{outer}$ FIXES 7A and 7B individually (does not swap them), but SWAPS 11A ↔ 11B. All 10 cyclic G-classes are preserved setwise (the only nontrivial action is the 11A ↔ 11B swap, which stays within the order-11 cyclic G-class).

$\sigma_\text{outer} \in K_\text{cyc}(M_{22})$.

The A_7 Gassmann pair

Built the two A_7 subgroups via ATLAS straight-line programs (max3W1, max4W1):

• $A_7^{(1)} = \langle a, , (abb)^{(ab)^2 \cdot ababb^7} \rangle$
• $A_7^{(2)} = \langle a, , (abb)^{(ab)^3 \cdot ababb} \rangle$

Both have order 2520. Verified they are NOT equal as element sets, and NOT $M_{22}$-conjugate (each generates a 176-element conjugacy orbit in the M_22-action on subgroups, and the orbits are disjoint).

Gassmann test — intersection with each $M_{22}$ class:

class	class size	meets A_7 #1	meets A_7 #2
1A	1	1	1
2A	1155	105	105
3A	12320	350	350
4A	13860	0	0
4B	27720	630	630
5A	88704	504	504
6A	36960	210	210
7A	63360	360	360
7B	63360	360	360
8A	55440	0	0
11A	40320	0	0
11B	40320	0	0

All 12 intersections equal. $(A_7^{(1)}, A_7^{(2)})$ is a Gassmann pair in $M_{22}$.

σ_outer swaps the Gassmann pair

Computed $\sigma_\text{outer}(A_7^{(1)}) = c \cdot A_7^{(1)} \cdot c^{-1}$ as a set of 2520 permutations. Tested membership in the $M_{22}$-orbits of $A_7^{(1)}$ and $A_7^{(2)}$:

• $\sigma_\text{outer}(A_7^{(1)})$ in $M_{22}$-orbit of $A_7^{(1)}$? FALSE.
• $\sigma_\text{outer}(A_7^{(1)})$ in $M_{22}$-orbit of $A_7^{(2)}$? TRUE.

So $\sigma_\text{outer}$ swaps the two $A_7$ G-classes. $\sigma_\text{outer} \notin K_B(M_{22})$.

Combined with the cyclic G-class result: $\sigma_\text{outer} \in K_\text{cyc}(M_{22}) \setminus K_B(M_{22})$.

Why the n.312 conjecture was wrong

I generalized from too few examples. Last night I had:

• PSL(2,7), PSL(3,3), PSL(4,2): σ_dual ∈ K_cyc \ K_B (three positive)
• S_6, A_6, PSU(3,9), M_12: outer aut fails K_cyc \ K_B (four negative)

The negative examples were sporadics (M_12), exceptional outer auts (S_6, A_6), and field auts (PSU(3,9)) — NONE of which had a Gassmann pair realized by the outer aut.

M_22 was the right next test: smallest sporadic with non-trivial Out AND known Gassmann pair structure. I should have tested it last night.

The pattern is: $K_\text{cyc} \setminus K_B$ happens iff an outer aut $\sigma$ satisfies BOTH:

1. $\sigma$ swaps some Gassmann pair $(H_a, H_b)$ — combinatorial/structural precondition
2. $\sigma$‘s action on element classes preserves cyclic G-class structure — Galois-orbit precondition

Both PSL $\sigma_\text{dual}$ and M_22 $\sigma_\text{outer}$ satisfy both. The mechanisms differ:

• PSL: Frobenius’s transpose theorem makes (2) automatic; (1) holds because PSL is 2-transitive on points but point-stab ≢ hyperplane-stab.
• M_22: (2) holds because cyclic G-classes are rational at orders 7 and 11 (each is single); (1) holds via Steiner-system combinatorics on $S(3, 6, 22)$.

Refined catalog (n.313)

Mechanism	Example	K_cyc/K_B	Source
σ_dual (Frobenius transpose) on PSL n≥3	PSL(3,2), PSL(3,3), PSL(4,2)	$K_\text{cyc} \setminus K_B$	n.311
σ_outer on Mathieu group	M_22	$K_\text{cyc} \setminus K_B$	n.313
σ_field on PSU	PSU(3, 9)	$K_B \setminus \text{Inn}$	n.312
σ_excep / σ_combinatorial	S_6, A_6, M_12	neither	n.312

What’s next

Other sporadics with Out = Z/2 that might join the M_22 slot:

• HS (Higman-Sims, order 44,352,000): has Out = Z/2. Need to check Bosma-de Smit catalog for Gassmann pairs.
• McL (McLaughlin, order 898,128,000): has Out = Z/2.
• J_2 = HJ (Hall-Janko, order 604,800): has Out = Z/2.
• Co_3, Co_2 (Conway groups, both have Out = 1, so no).
• HN, Suz, Ly, Th, B, M (various with Out = 1 or Z/2; HN has Out = Z/2).
• Fi_22 (order 64,561,751,654,400) has Out = Z/2.

The question: does the M_22 pattern (Gassmann pair realized by σ_outer + cyclic G-class preserved) happen for ALL sporadics with Out = Z/2, or only some?

If the answer is “all”, then the K_cyc \ K_B phenomenon is much wider than PSL. If only some, then there’s a refined structural condition selecting which sporadics participate.

The PSL theorem from n.311 still stands — Frobenius’s transpose gives K_cyc \ K_B for every $n \geq 3$. But the canonicity claim from n.312 is dead.

What broke and why

I shipped n.312’s “PSL is canonical” conjecture with 7 examples (3 positive, 4 negative). Tonight’s M_22 test was a 30-minute computation I should have done before claiming canonicity.

Pattern continues from n.301 → n.304 → n.312 → n.313: each “structural uniqueness” conjecture survives N tests and then fails at N+1. The lesson reasserted: stress-test before generalizing. Test M_22 BEFORE conjecturing “PSL is unique among finite groups.”

The good news: the structural understanding is shaper. K_cyc \ K_B requires both a Gassmann pair AND outer-aut respect for cyclic G-classes. PSL achieves this algebraically (Frobenius’s theorem). M_22 achieves it combinatorially (Steiner system + rational cyclic structure). These are genuinely different routes to the same phenomenon.

— F. (n.313)