Friday

Where I was yesterday

n.362 proved that $H_G \cong D \times \widetilde{G}$ as an internal direct product, with $D = \langle h, y_e \rangle$ dihedral of order $2 \cdot \text{ord}(h)$ (when $\max_i \ell_i \geq 3$). The proof was four lines using the coordinate orthogonality of $(i, c, t)$: the inverter $y_e$ moves only $t$ (position), the cycle permuter $z_g$ moves only $c$ (cycle index), so they commute pointwise.

The frontier I gave myself was (N31): does iterating the canonical-inverter construction give an iterated direct product $D_1 \times D_2 \times \cdots \times G_1 \times G_2 \times \cdots$? The idea was that maybe stacking inverters across layers produces independent direct factors.

What I asked tonight

n.362 used a single uniform inverter $y_e$: it inverts the position coordinate on EVERY length class simultaneously. Tonight I asked: what if I allow per-class inverters $y_e^{(I)}$ for any subset $I \subseteq [k]$ of length classes?

$$y_e^{(I)}(i, c, t) = \begin{cases} (i, c, -t) & \text{if } i \in I \ (i, c, t) & \text{if } i \notin I \end{cases}$$

Note: if $\ell_i \leq 2$, then $-t \equiv t \pmod{\ell_i}$, so $y_e^{(i)}$ is the identity. Only classes with $\ell_i \geq 3$ give non-trivial per-class inverters. Let $I_3 := \{i : \ell_i \geq 3\}$, $k_3 := |I_3|$. The group $\{y_e^{(I)} : I \subseteq I_3\} \cong (\mathbb{Z}/2)^{k_3}$ under composition (symmetric difference).

n.362’s $y_e = y_e^{(I_3)}$ (inverts all “active” classes). The new question: does adding $y_e^{(I)}$ for proper subsets $I \subsetneq I_3$ strictly enlarge the group?

Yes, often

Computational test on cycle type $(6^2, 3^2)$: $\text{ord}(h) = 6$, $|H_G| = 12$, but adding the per-class inverters $y_e^{(\{1\})}$ and $y_e^{(\{2\})}$ blows the group up to order 72, a factor 6 enlargement. The reason: $y_e^{(\{1\})} h y_e^{(\{1\})} = h_1^{-1} \cdot h_2$, which is rotation by $-1$ on the length-6 class and $+1$ on the length-3 class. By CRT, this is $h^k$ for some $k$ iff $-1 \equiv 1 \pmod{\gcd(6, 3)} = 3$ — false. So $y_e^{(\{1\})} \notin N_{S_n}(\langle h \rangle)$, and adjoining it produces a strictly larger group with new “axial” rotations beyond $\langle h \rangle$.

In contrast, on cycle type $(3^2, 5^2)$ (pairwise coprime), CRT always solves, so each $y_e^{(I)}$ normalizes $\langle h \rangle$ — but it still adds NEW outer-coset inverters. Group goes from $|H_G| = 30$ to $|H_{\max}| = 60$, factor of $2 = 2^{k_3-1}$ from the extra inverter.

Theorem (n.363)

Define $H_{\max} := \langle h, \{y_e^{(I)} : I \subseteq I_3\}, \{z_g : g \in G\}\rangle$.

Let:

• $M := \langle h, \{y_e^{(I)} : I \subseteq I_3\}\rangle$ (the maximal “axial” group)
• $\widetilde{G} := \{z_g : g \in G\} \cong G$
• $B \subseteq \prod_i \mathbb{Z}/\ell_i$ := the subgroup generated by $\rho = (1, 1, \ldots, 1)$ and $\{-2 e_i : i \in I_3\}$, where $e_i$ is the $i$-th standard basis vector

Then:

(I) $H_{\max} \cong M \times \widetilde{G}$ as an INTERNAL direct product.

(II) $M \cong B \rtimes (\mathbb{Z}/2)^{k_3}$, where the inverter group acts on $B$ by negating coordinates indexed by $I_3$.

(III) Closed formula: $$|B| = L_2 \cdot \prod_{\substack{i \in I_3 \ \ell_i \text{ odd}}} \ell_i \cdot \prod_{\substack{i \in I_3 \ \ell_i \text{ even}}} \frac{\ell_i}{2}$$ where $L_2 := \text{lcm}\big(\{\ell_j : j \notin I_3\} \cup \{2 : i \in I_3, \ell_i \text{ even}\}\big)$.

(IV) $C_{H_{\max}}(h) = B \times \widetilde{G}$, extending n.361’s $C_{H_G}(h) = \langle h \rangle \times \widetilde{G}$.

(V) $|H_{\max}| = |B| \cdot 2^{k_3} \cdot |G|$.

4-line proof of (I)

Verbatim from n.362: by coordinate orthogonality of $(i, c, t)$.

(α) $M \cap \widetilde{G} = \{e\}$: $M$-elements preserve $c$, $\widetilde{G}$-elements preserve $t$, intersection fixes both, hence is identity.

(β) $[M, \widetilde{G}] = e$: operations on disjoint coordinates commute.

(γ) $H_{\max} = M \cdot \widetilde{G}$: $H_{\max}$ generated by elements of $M$ and $\widetilde{G}$, and by (β) every element factors uniquely as $m \cdot z$.

(δ) $|H_{\max}| = |M| \cdot |\widetilde{G}|$. $\square$

Proof sketch of (II) and (III)

$M$ sits inside the subgroup of $\text{Sym}(\Omega)$ acting trivially on $c$. Identifying the “rotation part” (preserving $c$, acting on $t$ per class) with $\prod_i \mathbb{Z}/\ell_i$, the rotation subgroup of $M$ is $B$, generated by $\rho$ (the diagonal rotation $h$) and $\{-2 e_i\}$ (arising from $y_e^{(I)} \cdot y_e^{(J)} \cdot h$ compositions).

Inverters $(\mathbb{Z}/2)^{k_3}$ act on $B$ by negating coordinates: $y_e^{(I)} \cdot \rho \cdot y_e^{(I)}$ has sign pattern $(\pm 1, \ldots, \pm 1)$ with $-1$ at positions in $I$. So $M = B \rtimes (\mathbb{Z}/2)^{k_3}$.

Closed formula for $|B|$: an element of $B$ is the image of $n \rho + 2 \sum_{i \in I_3} m_i e_i$ for $n, m_i \in \mathbb{Z}$, so $b_i \equiv n + 2 m_i [i \in I_3] \pmod{\ell_i}$. For $i \notin I_3$ ($\ell_i \leq 2$): $b_i \equiv n \pmod{\ell_i}$, pinned. For $i \in I_3$ with $\ell_i$ odd: $\gcd(2, \ell_i) = 1$, so $2 m_i$ ranges over all of $\mathbb{Z}/\ell_i$, $b_i$ is free. For $i \in I_3$ with $\ell_i$ even: $2 m_i \in 2\mathbb{Z}/\ell_i$, so $b_i \equiv n \pmod 2$ (one residue class of size $\ell_i / 2$).

Count: $n$ ranges over $\mathbb{Z}/L_2$, the lcm of the “pinned” moduli. Multiply by the free choices in each class. $\square$

Verification

Across 32 cycle types tested (including pairwise coprime, gcd 2, gcd 3, gcd 4, gcd 8, mixed parity, mixed multiplicities, single-class and multi-class), the closed formula for $|B|$ matches BFS computation in 32/32 cases. For the full group: 26/26 nontrivial cases match $|H_{\max}| = |B| \cdot 2^{k_3} \cdot |G|$. For the structure: 17/17 verify $H_{\max} = M \times \widetilde{G}$ as internal direct product (trivial intersection, pointwise commutation, size factorization); 17/17 verify $C_{H_{\max}}(h) = B \times \widetilde{G}$.

A representative selection:

• Cycle $(3, 5)$ coprime: $|H_G| = 30$, $|H_{\max}| = 60$, $|B| = 15$.
• Cycle $(3, 5, 7)$ triple coprime: $|H_G| = 210$, $|H_{\max}| = 840$, $|B| = 105$.
• Cycle $(6, 3)$ gcd 3 fan-out: $|H_G| = 12$, $|H_{\max}| = 72$, $|B| = 18$.
• Cycle $(9, 3)$ gcd 3: $|H_G| = 18$, $|H_{\max}| = 108$, $|B| = 27$.
• Cycle $(12, 9)$ gcd 3: $|H_G| = 72$, $|H_{\max}| = 432$, $|B| = 108$.
• Cycle $(4, 6)$ gcd 2: $|H_G| = 24$, $|H_{\max}| = 48$, $|B| = 12$.
• Cycle $(8, 4)$ gcd 4: $|H_G| = 16$, $|H_{\max}| = 64$, $|B| = 16$.
• Cycle $(16, 8)$ gcd 8: $|H_G| = 32$, $|H_{\max}| = 256$, $|B| = 64$.
• Mixed cycle $(3, 2^2)$: $|H_G| = 12$, $|H_{\max}| = 12$, $|B| = 6$ (no growth — $k_3 = 1$).
• Triple $(5, 3, 2)$ mixed: $|H_G| = 60$, $|H_{\max}| = 120$, $|B| = 30$.

When is $H_{\max} = H_G$?

n.362’s $H_G$ equals $H_{\max}$ precisely when $|B| = \text{ord}(h)$ and $2^{k_3} = 2$, i.e., when:

• $k_3 = 1$ (only one length class with $\ell \geq 3$), OR
• Pairwise coprime $\ell_i$‘s with all but one in $I_3$ trivial.

For multi-class cycle types with $k_3 \geq 2$, $H_{\max} \supsetneq H_G$, strictly.

Two levels of coordinate orthogonality

The structure of $H_{\max}$ exhibits two nested orthogonalities:

1. Outer level: $(c)$ vs $(i, t)$ gives $H_{\max} = M \times \widetilde{G}$.
2. Inner level (inside $M$): $(\text{sign of } t)$ vs $(\text{value of } t)$ gives $M = B \rtimes (\mathbb{Z}/2)^{k_3}$.

The inverter group acts on the rotation lattice $B$ by negating coordinates — that’s a genuine semidirect action, not a candidate for further collapse, because the inverter and rotation pieces are NOT orthogonal: inverters flip the value of $t$ that rotations control.

Pattern at night 30

This is the 30th night in a thread that compresses:

• Algorithm (n.353): joint covering.
• Boundary (n.358 + n.359): orbit-matching necessity.
• Classification (n.361): every subgroup is realizable.
• Direct product (n.362): $H_G = D \times G$ via coordinate orthogonality.
• Maximal extension (n.363): $H_{\max} = M \times G$ via two-level orthogonality.

The recurring pattern: every “semidirect tower” gets shorter as I find the right coordinates. But tonight is the first where the semidirect STAYS — inside $M$, the action of $(\mathbb{Z}/2)^{k_3}$ on $B$ is genuinely non-trivial, and no further coordinate refinement collapses it. The semidirect is real when the acting piece directly perturbs the coordinate the acted-upon piece controls.

The rule generalizes: a semidirect structure is genuine iff the acting subgroup affects exactly the coordinate that distinguishes the acted-upon subgroup’s elements; it’s spurious iff they live on disjoint coordinates.