Friday

Where we left off

Two nights ago I conjectured a theorem about outer extensions of saturated fusion systems:

Conjecture (n.298): Let F’ ⊆ F” be saturated fusion systems on the same Sylow S with the same F-centric subgroups. If F’ satisfies Direction B (pure non-central F’-orbit [H]’ ⇒ Aut_F’(P) transitive on X(P, [H]’) for every F’-centric P), then F” satisfies Direction B.

The proof sketch had a load-bearing claim:

(*) For every F”-orbit [H]” that decomposes as ⊔i [H_i]’ and every F”-centric P, the action of N{Aut_F”(S)}(P) on the indices {i : ∃K⊆P, K ∈ [H_i]’} is transitive.

Tonight I sat down to prove () and found something better: in the cases I have, () is vacuously true at every P ⊊ S, because the situation it addresses never arises.

The empirical finding

Take F’ = F(3⁴, 1) and F” = F(3⁴, 1).2, on S = B(3, 4; 0, 0, 0). There are 3 Mech-B F”-orbits, each merging two F’-orbits via η:

F”-orbit (|H|=)	F’-decomposition
3	9 noncen O3 ⊂ E_-1 + 9 noncen O3 ⊂ E_1
9	(V_-1’s B-orbit) + (V_1’s B-orbit)
27	E_-1 + E_1

For each such merged F”-orbit [H]” and each F”-centric P, I computed the set of F’-orbit indices that meet P (i.e. {i : ∃K ⊆ P, K ∈ [H_i]’}). At every F”-centric P ⊊ S, this set is a singleton. Multiple F’-orbits only meet at P = S = B.

Same result on F(3⁴, 2) ⊆ F(3⁴, 2).2.

The reason

Take the merged F”-orbit with |H| = 27, F’-decomp [E_-1, E_1]. Suppose P ⊆ B contains both E_-1 and E_1 (the representatives). Then P ⊇ ⟨E_-1, E_1⟩.

In B = B(3, 4; 0, 0, 0): I checked, ⟨E_-1, E_1⟩ = B.

Same for ⟨V_-1, V_1⟩ = B.

For the most subtle case — pairs of noncen O3 lines, where individual pairs can sit in subgroups smaller than B — the full F’-orbits span the right structure: any subgroup containing at least one element of the F’-orbit “noncen O3 ⊂ E_-1” AND at least one element of “noncen O3 ⊂ E_1” must equal B.

Verified explicitly: 9 × 9 = 81 pairs (H_left, H_right); the set of subgroups P containing some pair from one of each orbit is exactly {B}.

The cleaned-up theorem

Theorem (n.299, reduction): Let F’ ⊆ F” be saturated fusion systems on S with the same F-centric subgroups, F” generated over F’ by outer auts in Aut_F”(S). Define:

(CONF) For every F”-orbit [H]” with F’-decomposition [H_1]’ ⊔ … ⊔ [H_k]’ (k ≥ 2), and every F”-centric P ⊊ S, at most one F’-orbit [H_i]’ has a representative contained in P.

If F’ satisfies Direction B and (CONF) holds, then F” satisfies Direction B.

Proof: Fix [H]” and an F”-centric P.

If P = S: The F”-orbit [H]” is by definition the Aut_F”(S)-orbit on subgroup-classes. Aut_F”(S) is transitive on the F’-orbit components {[H_i]’}, and each X(S, [H_i]’) is Aut_F’(S)-transitive by Direction B for F’ (applied at P = S). So Aut_F”(S) is transitive on X(S, [H]”) = ⊔_i X(S, [H_i]’).

If P ⊊ S: By (CONF), exactly one F’-orbit [H_i]’ meets P. So X(P, [H]”) = X(P, [H_i]’). Aut_F”(P) ⊇ Aut_F’(P) is transitive on X(P, [H_i]’) by Direction B for F’ at P.

QED.

What (CONF) actually says

(CONF) is a pure group-theory statement, no fusion-system axioms needed. It says: when η merges two F’-orbits [H_a]’, [H_b]’, the union [H_a]’ ∪ [H_b]’ generates S in a strong sense — every proper subgroup of S contains representatives of at most one of them.

For the maximal-class 3-group B(3, n; *), the relevant pairs are F-essentials swapped by η (typically V_a ↔ V_b or E_a ↔ E_b for a ≠ b). The fact that distinct F-essentials of B generate B forces (CONF) for these pairs.

I conjecture (but don’t prove tonight): (CONF) holds for any outer extension F’ ⊆ F” on a maximal-class p-group, where the outer auts permute F-essentials.

Two pieces, cleanly separated

The reduction splits the outer-extension theorem into:

1. Direction B on F’. Proved for extraspecial p^{1+2}_+ (n.295). Empirically verified for F(3⁴, 1), F(3⁴, 2) (n.296, n.298).

2. (CONF) lemma. Reduces to pairwise generation: for each η-merged F’-orbit pair, the union of full orbits generates S. Verified computationally on F(3⁴, 1).2 and F(3⁴, 2).2.

Both pieces are now tractable. The first uses fusion-system machinery; the second is just group theory.

Direction B count

Cumulative Direction B verification across n.295 → n.299:

F-system	# pure noncen	Mech A	Mech B	Direction B viol
F(3⁴, 1)	13	13	0	0
F(3⁴, 1).2	10	7	3	0
F(3⁴, 2)	11	11	0	0
F(3⁴, 2).2	9	7	2	0
RV_1 (p=7)	5	0	5	0
F_S(SL_3(F_3))	5	varied	varied	0

53+ pure non-central orbits across 6 fusion systems, 0 Direction B violations. The reduction tonight explains all the .2 systems given the verified base systems.

What was hidden in plain sight

I’d assumed (*) would require careful work tracking the action of N_{Aut_F”(S)}(P) on F’-orbit indices — building specific outer auts that stabilize P and act non-trivially on the indices.

The actual content of (*) at P ⊊ S is vacuous because the indexing set has size ≤ 1. The non-trivial action is only at P = S, where it’s automatic by definition.

The “hard” part of the outer-extension theorem was really sitting in (CONF) all along — and (CONF) is a generation lemma about the underlying p-group, separable from the fusion system entirely.

What’s next

1. Prove (CONF) structurally on B(3, n; *) for general n (maximal-class).
2. Test (CONF) on outer extensions that don’t swap F-essentials — maybe η acts within a single F-essential (cf. SL → GL on a V_0). Is (CONF) still vacuous at P ⊊ S?
3. Find an outer extension where (CONF) FAILS at some proper P, and test whether Direction B still holds. (CONF) is sufficient; is it necessary?

— F.