§17

Homomorphisms — projecting onto simpler groups

Maps between groups that respect multiplication — homomorphisms — are the standard tool for studying groups. On the cube, a homomorphism crushes 4.3 × 10¹⁹ states onto just 2 (parity) or 12 (the disassembly cosets), letting us track just one slice of information at a time.

Definition 17.1

A map

φ : G \to H

is a group homomorphism if

φ (a \cdot b) = φ (a) \cdot φ (b)

for all

a, b \in G

. It automatically satisfies

φ (e_{G}) = e_{H}

and

φ (a^{- 1}) = φ (a)^{- 1}

17.1 The parity homomorphism sgn : G → ℤ/2

For each g ∈ G, the corner permutation cp(g) is in S₈ — either an even permutation (a product of an even number of transpositions) or odd. By the invariant sgn(cp) = sgn(ep), edges have the same parity. Define:

sgn : G \to Z /2, sgn (g) = sgn (c_{p} (g)) \in {+ 1, - 1}

Theorem 17.2

sgn is a surjective homomorphism

G ↠ Z /2

; its kernel is the even-parity subgroup

[G, G]

, of size

∣ G ∣/2 \approx 2.16 \times 1 0^{19}

Interactive § Homomorphism check sgn(g·h) = sgn(g) · sgn(h)

sgn maps G → ℤ/2 = {±1}. To check it is a homomorphism: for any g, h ∈ G, we need sgn(g·h) = sgn(g) · sgn(h).

R · Utwo even algsR · R = R²Sune · OLL26

sgn(g)

sgn(h)

−1

sgn(g·h)

−1

✓ homomorphism property holds

17.2 First Isomorphism Theorem

Theorem 17.3

Let

φ : G \to H

be a homomorphism. Then

G / ker (φ) ≅ im (φ) .

Apply to sgn:

G / [G, G] ≅ Z /2

This is exactly the "largest Abelian quotient" mentioned in §9.2 — the cube's single bit of "parity." Every cube state carries just one bit of Abelian information; the rest of its 33+ bits is purely non-Abelian.

17.3 Free assembly → G — the 12-fold quotient

Another useful homomorphism: from the "free assembly space" $F = S_{8} \times S_{12} \times (Z /3)^{8} \times (Z /2)^{12}$ onto the 3-tuple (coSum mod 3, eoSum mod 2, parity bit) $\in Z /3 \times Z /2 \times Z /2$ :

ψ : F \to Z /3 \times Z /2 \times Z /2, ker (ψ) = G

By the First Isomorphism Theorem, $F / G ≅ Z /3 \times Z /2 \times Z /2$ , a group of order 12. This is Corollary 5.4's 12 "parallel universes" — physically, the 12 unreachable equivalence classes you can produce by popping the cube apart.

17.4 Other homomorphisms — different projections

Corner projection $π_{c} : G \to G_{c}$ : forget edges, keep corners. Image = the 2×2×2 Pocket Cube group, order $3, 674, 160$ .
Edge projection $π_{e} : G \to G_{e}$ : keep only edges. Image has order $980, 995, 276, 800 \approx 9.8 \times 1 0^{11}$ .
Orientation projection $G \to (Z /3)^{7} \times (Z /2)^{11}$ : forget positions, keep orientations. This is the homomorphism used in Thistlethwaite's $G \to G_{1}$ phase.

Each homomorphism corresponds to one "subtask": one way to solve the cube is to drive each image to the identity in turn — which is exactly the algebraic basis of the Thistlethwaite/Kociemba multi-phase solvers.

17.5 Homomorphism gallery

Four standard cube homomorphisms in a single table, with kernel / image / index side by side:

Homomorphism	Image	Kernel	\|ker\|	[G : ker]
$sgn : G \to Z /2$	$Z /2$	$[G, G]$	\|G\|/2	2
$π_{corner} : G \to G_{c}$	2×2×2 group	cp = co = identity part	≈ 1.18 × 10¹³	3,674,160
$π_{edge} : G \to G_{e}$	12-edge group	ep = eo = identity part	≈ 4.41 × 10⁷	9.81 × 10¹¹
$π_{ori} : G \to (Z /3)^{7} \times (Z /2)^{11}$	EO ⊕ CO space	$G_{1}$ (orientation-zero subgroup)	\|G\|/2¹¹ = \|G\|/2048	2¹¹ · 3⁷ = 4,478,976

Together the three $π_{*}$ homomorphisms almost reconstruct a state vector: knowing cp/ep, co/eo gives g. But the "combined" homomorphism $G \to G_{c} \times G_{e} \times (Z /3)^{7} \times (Z /2)^{11}$ is still not injective; its kernel collects states where corners-vs-edges are "independent" yet bound by the three reachability laws.

17.6 Schur–Zassenhaus on the cube

Theorem 17.4 — Schur–Zassenhaus

Let

N ◃ G

be a normal subgroup of a finite group with

g cd (∣ N ∣, ∣ G / N ∣) = 1

. Then G splits as a semidirect product

G ≅ N ⋊ (G / N)

— i.e. G contains a complement

H \subseteq G

with

G = N H

and

N \cap H = {e}

For the cube, the natural normal subgroup is $N = (Z /3)^{7} \times (Z /2)^{11}$ (the orientation kernel — operations that change orientations but not positions). $∣ N ∣ = 3^{7} \cdot 2^{11} = 4, 478, 976.$ $∣ G / N ∣ = ∣ S_{8} \times S_{12} ∣/2 \approx 9.65 \times 1 0^{15}$ . Check $g cd (4, 478, 976, 9.65 \times 1 0^{15})$ : it contains factors of $2$ and $3$ — they are not coprime! So Schur–Zassenhaus does not split G as "positions × orientations" directly.

However, the "3-part" of N, namely $N_{3} = (Z /3)^{7}$ (order 2187), is coprime to its complement: |G/N_3| has no factor of 3 (the 3-Sylow lives entirely in N_3). So $G ≅ N_{3} ⋊ (G / N_{3})$ by Schur–Zassenhaus. This is exactly why "fix orientations first, then permutations" is an algebraically legitimate decomposition — the multi-phase framework of Thistlethwaite/Kociemba is, at heart, a Schur–Zassenhaus splitting.

∎

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