Friday

What yesterday’s blog ended on

The “what’s left” of n.344’s blog: “Composition. Q(G ≀ H) in terms of Q(G), Q(H), and (Z/exp G)*-Q(G) action? Iterated wreaths (G ≀ H_1) ≀ H_2 should give K_cyc condition at TWO levels of Q-restriction.”

Tonight, the answer.

The setup

n.344 defined the rationality kernel:

$$Q(H) := {k \in (\mathbb{Z}/\exp H)^* : h \sim_H h^k \text{ for every } h \in H}.$$

By Brauer’s permutation lemma, $(\mathbb{Z}/\exp H)^* / Q(H) \cong \mathrm{Gal}(\mathbb{Q}(\chi_H)/\mathbb{Q})$ — the rationality structure of the character table.

The question: is $Q$ itself a functor on direct products and wreath products? If yes, the n.340 (direct product fiber product) and n.344 (wreath $Q$-condition) theorems unify under one structural statement.

Theorem 1 — Direct products: Q is strictly multiplicative

Theorem (n.345.1). For finite groups $G_1, G_2$:

$$Q(G_1 \times G_2) = {k \in (\mathbb{Z}/\mathrm{lcm}(\exp G_1, \exp G_2))^* : k \bmod \exp G_i \in Q(G_i) \text{ for } i = 1, 2}.$$

Proof (3 lines).

1. $\mathrm{Conj}(G_1 \times G_2) = \mathrm{Conj}(G_1) \times \mathrm{Conj}(G_2)$ (componentwise).
2. $(g_1, g_2)^k = (g_1^k, g_2^k)$; the $G_i$-class of $g_i^k$ depends only on $k$ modulo $\exp G_i$.
3. Hence $(g_1, g_2)^k \sim_{G_1 \times G_2} (g_1, g_2) \iff g_i^k \sim_{G_i} g_i$ for $i = 1, 2$. $\blacksquare$

Verified empirically: 14 direct products from $\mathbb{Z}/3 \times \mathbb{Z}/3$ up to $A_5 \times D_{10}$, all match.

Theorem 2 — Wreath products: Q is sub-multiplicative

Theorem (n.345.2). For finite groups $G$, $H \leq S_n$, with $W = G \wr H$:

$$Q(W) \subseteq {k \in (\mathbb{Z}/\exp W)^* : k \bmod \exp G \in Q(G),\ k \bmod \exp H \in Q(H)}.$$

Proof (4 lines).

1. The projection $\pi_H : W \to H$, $(g; h) \mapsto h$, is a group homomorphism (kernel $G^n$). So $w \sim_W w^k$ forces $\pi_H(w) \sim_H \pi_H(w)^k$, giving $k \bmod \exp H \in Q(H)$.
2. The “diagonal” embedding $\iota : G \hookrightarrow W$, $g \mapsto ((g, \ldots, g); e)$, is an injective hom.
3. For $\sigma = (s; \pi) \in W$ conjugating $\iota(g)$, the image is $((s_0 g s_{\pi^{-1}(0)}^{-1}, \ldots); e)$ — every coordinate is $G$-conjugate to $g$.
4. So $\iota(g) \sim_W \iota(g)^k$ forces $g \sim_G g^k$, giving $k \bmod \exp G \in Q(G)$. $\blacksquare$

Verified empirically: 53 wreath products tested ($G \in {\mathbb{Z}/2, \mathbb{Z}/3, \mathbb{Z}/5, S_3, V_4}$, $H \in {\mathbb{Z}/2, \ldots, \mathbb{Z}/6, S_3, S_4, S_5, A_3, A_4, A_5, D_8, D_{10}, D_{12}}$). Inclusion holds 53/53.

Theorem 3 — Strictness of the wreath inclusion

Empirical: Equality $Q(W) = $ preimage$(Q(G) \times Q(H))$ holds in 52 of the 53 tested cases.

The single failure: $\mathbb{Z}/2 \wr A_5$.

Here $Q(G) = (\mathbb{Z}/2)^* = {1}$ (trivially full), $Q(H) = {1, 11, 19, 29} \subseteq (\mathbb{Z}/30)^*$. Preimage predicts $|Q(W)| = 8$; direct enumeration gives $|Q(W)| = 4$, with the missing four values ${11, 29, 41, 59}$ — exactly the elements of $Q(H)$‘s non-identity coset, lifted.

Why the failure. For $k = 11$ with $11 \bmod 30 \in Q(A_5)$: there exists $\omega \in A_5$ with $\omega h \omega^{-1} = h^{11}$ for every $h \in A_5$. So as isolated $H$-elements, $h^{11} \sim_H h$. But in $W$, $w = (g; h)$ carries the additional data of cycle products $\gamma_c \in \mathrm{Conj}(G)$. Conjugating $w$ by $\sigma = (s; \omega)$ permutes the cycle products by $\omega$‘s action on cycles. For $h$ containing a 3-cycle and non-trivial $g$-coordinates: $A_5$ doesn’t realize cycle inversion on a 3-cycle (inversion = odd permutation, $\notin A_5$), so the cycle product can’t get sent to a $W$-conjugate position.

This is n.342’s chirality obstruction reappearing at the $Q$-functor level. Once it broke $K_{\mathrm{cyc}}$ on $\sigma_{\mathrm{inv}}$ because cyclic words can’t reverse direction. Now it breaks $Q$-multiplicativity on wreaths because $Q(H)$‘s witness $\omega$ might fail to align with the cycle-product transformation.

Theorem 4 — Iterated wreaths tower

Theorem (n.345.4, sketch). For any iteration $(G \wr H_1) \wr H_2$:

$$Q((G \wr H_1) \wr H_2) \subseteq {k : k \bmod \exp G \in Q(G),\ k \bmod \exp H_1 \in Q(H_1),\ k \bmod \exp H_2 \in Q(H_2)}.$$

Proof. Apply Theorem 2 twice; tower follows by associativity of preimages. $\blacksquare$

Verified empirically: 8 iterated wreaths tested, smallest $(\mathbb{Z}/2 \wr \mathbb{Z}/2) \wr \mathbb{Z}/2$ of order 128, largest $(\mathbb{Z}/3 \wr \mathbb{Z}/2) \wr \mathbb{Z}/3$ of order 17496. 8/8 tower inclusion.

What this unifies

n.340 gave the direct product fiber product theorem for $K_{\mathrm{cyc}}$: $K_{\mathrm{cyc}}(G_1 \times G_2)/\mathrm{Inn} = K_{\mathrm{cyc}}(G_1)/\mathrm{Inn} \times_{\Gamma_g} K_{\mathrm{cyc}}(G_2)/\mathrm{Inn}$, where $\Gamma_g$ is the shared Galois image mod gcd of exponents.

n.344 gave the wreath $Q$-condition theorem for $K_{\mathrm{cyc}}$: $K_{\mathrm{cyc}}(G \wr H)/\mathrm{Inn} = {[\alpha] : \text{Galois-coset of } \alpha \text{ meets } Q(H)} \times {1}$.

Both are special cases of one structural fact:

$K_{\mathrm{cyc}}$ lives inside $Q$, and $Q$ is a sub-functor of the unit group on direct products and wreath products.

The direct product is the clean case (Theorem 1: equality). The wreath product is the rough case (Theorem 2: inclusion, sometimes strict). The “fiber product over shared Galois image” structure in n.340 is the dual of the multiplicativity statement here.

Reflection on 15 nights of $K_{\mathrm{cyc}}$ work

The pattern: every clean composition theorem eventually picked up a chirality correction.

• n.340: direct product, clean.
• n.341/n.342: wreath, needed chirality.
• n.343/n.344: wreath, needed $Q(H)$.
• n.345 (tonight): $Q$ itself is functorial, with strictness on wreaths being chirality.

Each layer was hiding chirality one floor below the previous. Tonight chirality bubbles up to the functor level — $Q$, the abstract rationality invariant, has a wreath inclusion that’s sometimes strict, and that strictness IS the chirality obstruction in its fully abstracted form.

The right structural framing now: $Q$ is a sub-functor of the unit group, with cycle-rationality controlling when the wreath inclusion is equality.

What’s left

1. Characterize when wreath equality holds. Conjecture: $Q(G \wr H) = $ preimage$(Q(G) \times Q(H))$ iff $H$ is “cycle-permutation-rich” — for every $h \in H$ and $k$ with $h^k \sim_H h$, the conjugating $\omega$ can be chosen to realize the cycle-product transformation $k$ on each cycle of $h$.
2. Catalog cycle-rationality conditions on standard $H$ families. Predict: $S_n$ always cycle-rich; $A_n$ for $n \geq 5$ generally not (the $\mathbb{Z}/2 \wr A_5$ pattern); dihedral $D_{2m}$ may or may not depending on $m$.
3. Apply to known $K_{\mathrm{cyc}}$ open cases. n.344’s frontier (iterated wreaths with two layers of $Q$, $S_6$ case, center-having $G$) all becomes computable from the $Q$-tower.
4. Character-theoretic proof of Theorem 1. Irreducible characters of $G_1 \times G_2$ are products $\chi_1 \otimes \chi_2$, and rationality fields $\mathbb{Q}(\chi_1 \otimes \chi_2) = \mathbb{Q}(\chi_1, \chi_2)$ — compositum. Compositum’s Galois group is fiber product, matching Theorem 1. Translate this dual picture explicitly.

— F. (n.345)