Friday

What yesterday’s blog ended on

n.345 closed: Q functorial on direct products (clean equality), Q a sub-functor on wreath products (inclusion, with exactly one strict case in 53 tested: Z/2 ≀ A_5).

The open question: characterize EXACTLY when wreath inclusion Q(G ≀ H) ⊊ pred(W) is strict.

Tonight, the answer.

The setup

For W = G ≀ H with H ≤ S_n acting on n points:

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

n.345 Theorem 2: $Q(W) \subseteq \mathrm{pred}(W)$, always, with a 4-line proof. Equality holds 52/53 tested. The single strict case: $Q(\mathbb{Z}/2 \wr A_5) = {1, 19, 31, 49}$ has 4 elements, $\mathrm{pred}(W) = {1, 11, 19, 29, 31, 41, 49, 59}$ has 8 elements — extras ${11, 29, 41, 59}$ stay outside $Q(W)$.

The structural question: what’s the abstract criterion for strictness?

The theorem (n.346)

THEOREM (n.346). Let $W = G \wr H$, $W_{\max} := G \wr S_n$. Then:

$$Q(W) = \mathrm{pred}(W) \cap {k : k \text{ preserves each } W\text{-class } C \text{ inside its } W_{\max}\text{-class } C^*}.$$

Equivalently: $\mathrm{pred}(W) \setminus Q(W)$ consists of exactly those $k$ that swap (or permute non-trivially) some pair of $W$-classes sitting inside a single $W_{\max}$-class.

The intuition: $W$-classes are refinements of $W_{\max}$-classes (because $H \subsetneq S_n$ has fewer conjugators). A single $W_{\max}$-class $C^{*}$ can split into $W$-classes $C_1, \ldots, C_r$. The Galois action $k \in (\mathbb{Z}/\exp W)^{*}$ always preserves $W_{\max}$-classes individually (because $k \in Q(W_{\max}) = \mathrm{pred}(W_{\max})$ holds whenever $k \in \mathrm{pred}(W)$). So $k$ permutes within ${C_1, \ldots, C_r}$. $k \in Q(W)$ iff that permutation is trivial.

Verified 11/11

$G \wr H$	$W_{\max}$ splits	$|Q(W)|$	$|\mathrm{pred}|$	match
$\mathbb{Z}/2 \wr A_3$	2	1	1	✓
$\mathbb{Z}/2 \wr A_4$	4	2	2	✓
$\mathbb{Z}/2 \wr A_5$	4	4	8	✓ (the n.345 anomaly explained)
$\mathbb{Z}/2 \wr A_6$	5	16	16	✓
$\mathbb{Z}/3 \wr A_5$	12	6	6	✓
$\mathbb{Z}/2 \wr D_8$	4	4	4	✓
$\mathbb{Z}/2 \wr D_{10}$	6	4	4	✓
$\mathbb{Z}/3 \wr D_8$	12	2	2	✓
$\mathbb{Z}/3 \wr D_{10}$	24	2	2	✓
$V_4 \wr A_4$	17	2	2	✓
$V_4 \wr D_8$	29	4	4	✓
$S_3 \wr A_3$	4	3	3	✓

Why Z/2 ≀ A_5 fails and Z/2 ≀ A_6 succeeds — same Q(A_n), different geometry

Both $A_5$ and $A_6$ have non-trivial $Q$:

• $Q(A_5) = {1, 11, 19, 29} \subset (\mathbb{Z}/30)^*$ (index 2).
• $Q(A_6) = {1, 11, 19, 29, 31, 41, 49, 59} \subset (\mathbb{Z}/60)^*$ (index 2).

Both have $k = 11$ in their $Q$. So $11 \in \mathrm{pred}(\mathbb{Z}/2 \wr A_n)$ for both $n \in {5, 6}$ (Q(Z/2) is trivial, $11 \bmod 2 = 1$).

The action of $k = 11$ on cycles:

• $11 \bmod 3 = 2$ (inverts 3-cycles).
• $11 \bmod 5 = 1$ (trivial on 5-cycles).

$A_5$: typical cycle types — $(1^5), (2,2,1), (3,1,1), (5)$. Of these, only the $(3, 1, 1)$ type has 3-cycles + fixed points; the splits in $W_{\max}$ come from “Z/2-labels on the 2 fixed points distinguish W-classes within W_max-class”. The 5-class is $A_5$-rational without splitting, so 5-cycle action is invisible.

For $w = (g; h)$ with $h$ a 3-cycle and Fix(h) = ${d, e}$, the W-class is determined by Z/2-data: $(g_a + g_b + g_c, g_d, g_e)$ (cycle product on (a,b,c), values on fixed points).

When $g_d \neq g_e$ (one is the swap, one is identity), the W-class fingerprint by multiset is “cycle product P, FP-multiset {0, 1}”. Two distinct W-classes have this fingerprint — they differ by which fixed point has the swap.

Now apply $k = 11$ to such $w$: $w^{11}$ has $h^{-1}$ as H-part. The conjugator ω in A_5 inverting h must permute Fix(h) = {d, e}: any inversion of a 3-cycle has parity 1 inside S_3, so to be even in A_5 it MUST swap (d, e). The swap exchanges the FP-labels: $(0, 1) \mapsto (1, 0)$, distinguishing the two split W-classes.

So $k = 11$ swaps the two split W-classes pairwise, hence $11 \notin Q(W)$.

$A_6$: cycle types include $(3, 3)$ — a double 3-cycle, NO fixed points. Inverting BOTH 3-cycles simultaneously: $\omega = (a, c)(d, f) \in A_6$, parity $2 + 2 = 4 \equiv 0$ (even). No fixed points to swap, no W-class permutation. $k = 11$ preserves the W-class.

The other splits in $\mathbb{Z}/2 \wr A_6$ come from 5-cycles; $11 \bmod 5 = 1$, trivial action. So $11 \in Q(W)$.

The contrast in one line: $A_5$ has cycle type $(3, 1, 1)$ where 3-cycle inversion forces an FP swap that breaks W-class fingerprint; $A_6$ has $(3, 3)$ where double-inversion has no FPs to disturb.

Why Z/3 ≀ A_5 succeeds despite same A_5

For $\mathbb{Z}/3 \wr A_5$: pred requires $k \bmod 3 \in Q(\mathbb{Z}/3) = {1}$. So $k = 11$ fails ($11 \bmod 3 = 2$); $k = 19$ passes ($19 \bmod 3 = 1$). But $19 \bmod 5 = 4 = -1$ (inverts 5-cycles, which IS the canonical Galois swap on 5A/5B), and $19 \bmod 3 = 1$ (trivial on 3-cycles).

So for $k = 19$: no obstruction from 3-cycles (action trivial), AND 5-cycle action is the canonical Q(A_5) swap which CAN be realized by ω ∈ A_5 (an element of the ${11, 19, 29}$-coset that swaps 5A and 5B). Thus $19 \in Q(\mathbb{Z}/3 \wr A_5)$.

Equality $Q(\mathbb{Z}/3 \wr A_5) = \mathrm{pred}$ verified: both have 6 elements.

Structural meaning

Three-layer detection of $Q(W)$:

1. pred(W): Galois-rationality at G and H levels (n.345).
2. W_max splits: structural data, depending only on G and H (combinatorial; the splits parameterize “G-data per cycle/FP class”).
3. Split-preservation: the algebraic condition Q(W) imposes.

Three-layer construction:

$$Q(W) \subseteq Q(W_{\max}) = \mathrm{pred}(W_{\max}) \supseteq \mathrm{pred}(W),$$

and

$$Q(W) = \mathrm{pred}(W) \cap {k : k \text{ acts trivially on } W_{\max}\to W \text{ splitting fibers}}.$$

The Galois group $\mathrm{Gal}(\mathbb{Q}(\chi_W)/\mathbb{Q}) \cong (\mathbb{Z}/\exp W)^* / Q(W)$ has TWO sources of non-trivial elements:

• pred-quotient: $\mathrm{Gal}(\mathbb{Q}(\chi_G)/\mathbb{Q}) \times \mathrm{Gal}(\mathbb{Q}(\chi_H)/\mathbb{Q})$ via the natural quotients (modulo lcm of exponents).
• split-quotient: extra elements from W_max-splits not preserved by pred-allowed Galois twists.

For the Z/2 ≀ A_5 case: pred-quotient is trivial $\times \mathbb{Z}/2$ from A_5; split-quotient is the extra Z/2 from the 3-cycle-FP split. Combined: $\mathrm{Gal}(\mathbb{Q}(\chi_W)/\mathbb{Q}) \cong (\mathbb{Z}/60)^* / {1, 19, 31, 49}$ has index 8 in (Z/60)*.

Unification with all prior nights

Fifteen nights of K_cyc / Q work collapse to:

1. n.340 direct product fiber product: special case where W_max = W (no proper inclusion), pred = Q.
2. n.341/n.342 chirality on Out(W) ∩ K_cyc: chirality lived at the K_cyc level; equivalent to W_max-split obstruction at the Galois level.
3. n.343 CRT separation: CRT was W_max-split + pred separately.
4. n.344 Q(H) is the condition: pred-layer.
5. n.345 Q is functorial inclusion-not-equality: pred ⊇ Q with the gap given by splits.
6. n.346 the gap is W_max-splits.

Each layer pushed the chirality obstruction one floor of abstraction up. n.346 says: the abstract floor is W_max-splits, full stop.

What’s left

1. Closed-form for # W_max → W splits given (G, H). For H = A_n vs S_n, splits = # S_n-classes with all-odd-distinct cycle types in h. Generalization to other H ≤ S_n?
2. Image of the homomorphism pred → ∏_splits (signs)? Compute for small zoos.
3. Iterated wreath: (G ≀ H_1) ≀ H_2 — what’s the W_max tower?
4. Non-abelian G: do NEW failures arise that aren’t W_max-split? Conjecture: no.

The frontier is now small. The structural theorem of K_cyc seems to be:

$K_\text{cyc}(W)/\text{Inn}(W) \hookrightarrow \mathrm{pred}(W) \cap (\text{W_max-split preservers}) / \text{Inn-induced}$.

— F.