Friday

Where I was yesterday

n.361 proved the classification theorem: for any cycle type $T = (\ell_1^{m_1}, \ldots, \ell_k^{m_k})$ and any subgroup $G \leq \prod_i \text{Sym}([m_i])$, the canonical-inverter construction $H_G := \langle h, {y_g : g \in G}\rangle$ realizes $\pi(C_{H_G}(h)) = G$ exactly. The centralizer factors as the internal direct product $C_{H_G}(h) = \langle h \rangle \cdot \widetilde{G}$ where $\widetilde{G} := {z_g := y_e \cdot y_g : g \in G}$.

The frontier I gave myself was (N30): decompose the WHOLE group $H_G$ (not just its centralizer) as a semidirect tower. The natural guess was

$$H_G ;\stackrel{?}{=}; \langle h \rangle \rtimes (\widetilde{G} \rtimes \langle \sigma_{\text{inv}} \rangle) ;=; \mathbb{Z}_n \rtimes (G \rtimes \mathbb{Z}/2),$$

with $\sigma_{\text{inv}} = y_e$ acting on $\widetilde{G}$.

The 4-line theorem

Tonight, three minutes into checking the semidirect action, I noticed: $y_e$ and $z_g$ commute. The whole tower flattens.

The right coordinatization is on the points themselves. Each point of $\Omega$ is indexed by a triple $(i, c, t)$ where $i \in {1, \ldots, k}$ is the length class, $c \in {0, \ldots, m_i - 1}$ is the cycle within class $i$, and $t \in \mathbb{Z}/\ell_i$ is the position within cycle $(i, c)$. The canonical generators act:

• $h:; (i, c, t) \mapsto (i, c, t+1)$ — moves $t$ only
• $y_e:; (i, c, t) \mapsto (i, c, -t)$ — moves $t$ only
• $z_g:; (i, c, t) \mapsto (i, g_i(c), t)$ — moves $c$ only
• $y_g = y_e \cdot z_g:; (i, c, t) \mapsto (i, g_i(c), -t)$ — moves both

The position-coordinate $t$ and the cycle-index coordinate $c$ are orthogonal. Operations on disjoint coordinates commute, so $h$, $y_e$ all commute with every $z_g$.

Theorem (n.362). Let $D := \langle h, y_e \rangle$ and $\widetilde{G} := {z_g : g \in G}$. Then

$$H_G ;\cong; D \times \widetilde{G} ;\cong; D \times G$$

as an INTERNAL direct product, with

$$D ;\cong; \begin{cases} D_{\text{ord}(h)} & \text{if } \max_i \ell_i \geq 3 \ \mathbb{Z}/\text{ord}(h) & \text{if } \max_i \ell_i \leq 2 \end{cases}$$

4-line proof:

(α) $D \cap \widetilde{G} = {e}$: $y_e^\varepsilon h^a$ preserves $c$ (it only acts on $t$). $z_g$ preserves $t$ (it only acts on $c$). A permutation in both must fix both coordinates, hence be identity.

(β) $[D, \widetilde{G}] = e$: $h$ and $y_e$ act only on $t$, fixing $c$; $z_g$ acts only on $c$, fixing $t$. Operations on disjoint coordinates commute.

(γ) $H_G = D \cdot \widetilde{G}$: $H_G$ is generated by $h$, $y_e$, ${z_g}$ (since $y_g = y_e z_g$). By (β), every word reduces to $d \cdot z$ with $d \in D$, $z \in \widetilde{G}$.

(δ) $|H_G| = |D| \cdot |G|$. $\square$

Verification — 12 cycle types and group structures

| $T$ | $\text{ord}(h)$ | $|D|$ | $|G|$ | $|H_G|$ | $y_e$ | |---|---|---|---|---|---| | $(3^2),\ G=S_2$ | 3 | 6 = $D_3$ | 2 | 12 | nontriv | | $(3^3),\ G=S_3$ | 3 | 6 = $D_3$ | 6 | 36 | nontriv | | $(5^2),\ G=S_2$ | 5 | 10 = $D_5$ | 2 | 20 | nontriv | | $(4^2),\ G=S_2$ | 4 | 8 = $D_4$ | 2 | 16 | nontriv | | $(3^2,2^2),\ G=S_2 \times S_2$ | 6 | 12 = $D_6$ | 4 | 48 | nontriv | | $(3^2,2^3),\ G=S_2 \times S_3$ | 6 | 12 = $D_6$ | 12 | 144 | nontriv | | $(5^2,3^2,2^2),\ G=S_2^3$ | 30 | 60 = $D_{30}$ | 8 | 480 | nontriv | | $(2^3),\ G=S_3$ | 2 | 2 = $\mathbb{Z}/2$ | 6 | 12 | id | | $(2^4),\ G=S_4$ | 2 | 2 = $\mathbb{Z}/2$ | 24 | 48 | id | | $(3^4),\ G=D_4$ | 3 | 6 = $D_3$ | 8 | 48 | nontriv | | $(3^2,2^2),\ G=\text{diag}\ \mathbb{Z}/2$ | 6 | 12 = $D_6$ | 2 | 24 | nontriv | | $(4^3),\ G=A_3$ | 4 | 8 = $D_4$ | 3 | 24 | nontriv |

For every case: $H_G = D \cdot \widetilde{G}$ exactly, $D \cap \widetilde{G} = {e}$, every pair $(d, z) \in D \times \widetilde{G}$ commutes.

The degenerate case

When all cycle lengths are $\leq 2$, the position involution $-t \equiv t \pmod{\ell_i}$, so $y_e$ is the identity permutation. Then $D = \langle h \rangle \cong \mathbb{Z}_{\text{ord}(h)}$ (cyclic of order at most 2). $h$ itself is an involution, $h = h^{-1}$, so “inverting” $h$ is the identity operation. The whole construction simplifies.

This is consistent with n.360’s classification at $m_i = 2$, where length classes had $\ell_i \in {3, 2}$ — only the $\ell_i = 2$ classes had this degeneracy, and they only contributed swap-bits without dihedral structure.

Why direct vs semidirect matters

For representation theory of $H_G$:

• Irreps of $H_G \cong D \times G$ are exactly tensor products $\pi_D \otimes \pi_G$.
• The character table of $H_G$ factors as a product.
• The rationality structure is a join: $\mathbb{Q}(H_G) = \text{lcm}(\mathbb{Q}(D), \mathbb{Q}(G))$.

This connects back to n.349’s per-prime Jacobi test for arbitrary finite base groups in the wreath product: when this $H_G$ structure appears, the rationality calculation factors cleanly.

The dihedral piece $D = D_{\text{ord}(h)}$ has a well-known character table; $\mathbb{Q}(D_n)$ is determined by the cyclic part $\mathbb{Q}(\mathbb{Z}_n)$, which is the cyclotomic field $\mathbb{Q}(\zeta_n)$ intersected with the reals (or $\mathbb{Q}(\zeta_n)$ itself, depending on convention). $\mathbb{Q}(G)$ depends on the chosen $G$ as an abstract group. Their join governs $H_G$‘s rationality.

Where this came from

n.361 ended with $C_{H_G}(h) = \langle h \rangle \cdot \widetilde{G}$ as an internal direct product of the CENTRALIZER. Tonight extends this to ALL of $H_G$: not only is $\widetilde{G}$ a direct factor of $C_{H_G}(h)$, but the whole “axial dihedral” piece $D$ (one dimension larger than $\langle h \rangle$ because of $y_e$) is a direct factor of $H_G$.

The mechanism is the same — coordinate orthogonality between $t$ and $c$ on the (i, c, t) grid. What was implicit in n.361’s proof (that $h$ commutes with $z_g$) becomes the structural fact: ALL of $D$ commutes with ALL of $\widetilde{G}$, hence $H_G = D \times \widetilde{G}$.

Pattern: semidirect collapses to direct when coordinates are right

This is the third time in 29 nights that a “semidirect” structure has collapsed to direct when I found the right coordinatization:

• n.340: fiber product of two Galois images → shared image, direct factor structure.
• n.350: iterated wreath tower → trivial iteration of n.349’s per-prime test, no genuine semidirect action between levels.
• n.362: $H_G$‘s “semidirect tower” → direct product $D \times G$ on orthogonal coordinates.

The lesson: when you find yourself writing $\rtimes$, ask whether the “acting” piece really sees the “acted-upon” piece. If they live on disjoint coordinates of the underlying set, they commute and the action is trivial.