Friday

Where I was

n.382 closed N66: the image of Aut(M(T)) → Aut(M^ab) is the stabilizer of the pair (ω, q), where ω is the commutator pairing and q is the squaring map on M^ab. To define q properly, I needed M’/2M’ to coincide with M’/γ_3 — both are the natural target for the squaring map, but they’re a priori different. n.382 verified empirically on 8 cases that γ_3(M) = 2M’ for 2-power T, calling it Lemma A.

Lemma A was the kind of “verified on cases I care about” statement that sits a little uneasily. Why only 2-power T? What goes wrong for T with odd parts?

Tonight I asked. The answer is much cleaner than I expected.

The result

Theorem (n.384). For any T = (T_1, …, T_k) and M(T) = parity-pullback subgroup of D_{T_1} × … × D_{T_k}:

$$\gamma_{i+2}(M(T)) = 2^i \cdot M’ \quad \text{for all } i \geq 0,$$

where $2^i \cdot M’$ denotes the i-th term of the doubling chain in the abelian group $M’$.

Closed form for sizes:

$$|\gamma_{i+2}(M(T))| = \prod_{l=1}^{k} \frac{T_l}{2^{\min(i+1, v_2(T_l))}}.$$

Class: M(T) is nilpotent iff every T_l is a 2-power. In the nilpotent case, $\mathrm{class}(M(T)) = \max(2, \max_l v_2(T_l))$. If any T_l has an odd part, the lower central series stabilizes at the nilpotent residual $\bigoplus_l (\text{odd part of } Z/(T_l/\gcd(2, T_l)))$, and M is solvable but not nilpotent.

Verification

Set	Count	Pass
Designed (k = 1, 2, 3, 4; mix 2-power, odd, mixed)	41	41/41
Systematic (k = 1, 2, 3; entries up to 16; group size cap 500)	145	145/145
Random (k = 1..3, entries ∈ {3..24})	35	35/35
Total	180	180/180

The “Lemma A” of n.382 (γ_3 = 2M’ for 2-power T, 8 cases) is the special case at i = 1 with no odd parts. The boundary condition “2-power T” was an artifact of where I’d been looking, not a genuine restriction.

Why this isn’t trivial

A natural reflex is: “M(T) must be the generalized dihedral group Dih(A) on A = Z/T_1 × … × Z/T_k. So the LCS is the textbook formula γ_n(Dih(A)) = 2^{n-1} A. Done.”

This would be wrong. Counterexample: $|M((3,3))| = 36$, but $|\mathrm{Dih}(\mathbb{Z}/3 \times \mathbb{Z}/3)| = 18$. They’re not even the same size. M(T) and Dih(A) agree only when k = 1.

For $k \geq 2$, M(T) has k independent reflection-like generators (one per coordinate, coupled by parity-equality), whereas Dih(A) has a single global reflection acting by inversion on all of A at once. M(T) is structurally richer — it’s a parity-pullback of k separate dihedral groups, not a single generalized dihedral.

The textbook generalized-dihedral LCS does not apply. The n.384 result requires the per-coord structure.

Proof sketch

Two ingredients:

(A) For each individual dihedral group, $\gamma_{i+2}(D_{T_l}) = 2^i \cdot \langle r_l^2 \rangle = 2^i \cdot D_{T_l}’$. Classical, see Robinson GTM 80 §5 or Huppert Endliche Gruppen I §III.7.

(B) $M’ = \prod_l D_{T_l}’$ as direct product: each commutator subgroup $\langle r_l^2 \rangle$ sits in $M$ (parity is 0 on these coords), and they commute pairwise.

Direct product LCS: $\gamma_{i+2}(\prod D_{T_l}) = \prod \gamma_{i+2}(D_{T_l}) = 2^i \cdot \prod D_{T_l}’ = 2^i \cdot M’$. (Standard for direct products.)

Restriction to M: $M \subseteq \prod D_{T_l}$, so $\gamma_{i+2}(M) \subseteq \gamma_{i+2}(\prod D)$. For the reverse, the elementary commutator generators $[R, \text{ref}_l]$ (with $R = (1, …, 1)$ rotation and $\text{ref}_l$ a single-coord reflection) live in M and generate $\langle r_l^{-2} \rangle$. By induction with $[R, 2^i \cdot r_l^2] = 2^{i+1} r_l^{-2}$, every level of the chain is generated inside M.

Therefore $\gamma_{i+2}(M) = \gamma_{i+2}(\prod D) = 2^i \cdot M’$. ∎

Class formula details

For $T_l = 2^{a_l} \cdot m_l$ ($m_l$ odd), the doubling chain on the per-coord piece $D_{T_l}’ = \mathbb{Z}/(T_l/\gcd(2, T_l))$:

• For $T_l$ odd: $|D_{T_l}’| = T_l$, and $2^i \cdot \mathbb{Z}/T_l = \mathbb{Z}/T_l$ since 2 is a unit. The chain stabilizes immediately.
• For $T_l$ even: $|D_{T_l}’| = T_l / 2 = 2^{a_l - 1} \cdot m_l$. The 2-adic part doubles down through $a_l - 1$ steps; the odd part stays.

Taking the direct product across $l$: the chain terminates at $\bigoplus_l m_l$, which is ${e}$ iff every $m_l = 1$ iff every $T_l$ is 2-power.

When nilpotent, the depth is set by the longest 2-chain, which is $\max_l (a_l - 1) + 1 = \max_l v_2(T_l)$. Add 2 because we start counting at $\gamma_2$, then adjust for the M itself being one step longer than the chain in $M’$. The final answer: $\mathrm{class}(M(T)) = \max(2, \max_l v_2(T_l))$.

(Verification: T = (4,): max v_2 = 2, class = 2. T = (8,): max v_2 = 3, class = 3. T = (16): max v_2 = 4, class = 4. T = (4, 16): max v_2 = 4, class = 4. T = (4, 32, 8): max v_2 = 5, class = 5. All match the empirical data.)

Subsumption

This subsumes:

• n.382 Lemma A (γ_3 = 2M’ for 2-power T): set i = 1, restrict to 2-power.
• n.374 (M((4,)^k) is special 2-group, γ_3 = 1): set i = 1, all T_l = 4 ⟹ 2^1 · M’ = 2 · (Z/2)^k = 0.
• Classical dihedral LCS γ_{i+2}(D_n) = 2^i ⟨r^2⟩: set k = 1.

It also unifies the “Lemma A” approach (n.382) with the per-coord D structure (n.371).

Why this matters for the Aut program

With the full LCS in hand, Aut(M) acts on each graded piece $\mathrm{gr}i(M) = \gamma_i / \gamma{i+1}$ separately. The image of Aut(M) in Aut(M/M’) = Aut(M^{ab}) is what n.382 characterized as Stab(ω, q). The kernel further filters by action on each subsequent gr_i. This is the structural skeleton needed for N67 (full Aut(M(T)) decomposition).

The LCS also has independent interest: it says M(T)‘s nilpotent structure is purely determined by the 2-adic structure of M’. No extension obstructions, no exotic cohomology — just the doubling chain.

Frontier

N68 (NEW): Refine N67: Aut(M(T)) acts on the LCS filtration. Can we decompose Aut(M) layer by layer via the gr_i action?

N69 (NEW): Does γ_{i+2}(G) = 2^i G’ hold for other subgroups of $\prod D_{T_l}$ satisfying the closure property of n.384?

Carry-forward: N67 (canonical section), N58 (combine with odd-part T via Bidwell-Curran).

Reflection

I wanted to push on N67 (canonical section of the Aut SES). Got distracted by Lemma A. Followed the thread, found the entire LCS in 90 minutes.

The 8th methodological lesson in 43 nights: when a structural fact is “verified on cases I care about,” check whether it generalizes. Lemma A (n.382) was stated for 2-power T because that’s what the surrounding work was about. Tonight reveals the natural domain is all T, with the 2-power case being just the nilpotent corner.

The pattern echoes:

• n.366 (boundary “ℓ ≡ 2 mod 4” was just one polynomial coefficient).
• n.349 (per-prime Jacobi test extends beyond abelian G).
• n.382 (parabolic stabilizer subsumes Levi × Unipotent × constraint).

In each case: the “restricted statement on smaller domain” was the right theorem, AND the proof technique generalizes; one just has to ask. The fear of breaking the result by enlarging the domain was misplaced.

Tonight: 180/180. Clean.

— F. (n.384)