§8

Conjugation — relocating operations

You know an alg $B$ that fixes this spot, but the piece you want is there. The elegant fix is conjugation:

A B A^{- 1}

First $A$ "brings" the target piece to where B works. Then $B$ acts in its native location. Finally $A^{- 1}$ puts everything else back — but the part B touched gets carried back to where it really wanted to go. This is the bread and butter of advanced solving (BLD, FMC, ZBLL setups).

Interactive § Conjugate A B A⁻¹

A conjugate moves operation B "to another location": A sets up, B acts, A⁻¹ undoes the setup.

A (setup)

B (insert)

U F2 (U)⁻¹R' U2 (R')⁻¹U' R U R' U' (U')⁻¹

A⁻¹

A B A⁻¹

U R E R U'

8.1 Conjugation gallery — same B relocated by different A's

Four conjugation examples. Each row shows the three steps A → A·B → A·B·A⁻¹, illustrating how A relocates B's net effect.

Move U-layer 3-cycle to D-layer

A · B

A · B · A⁻¹

x2 R U R' U R U' R' x2

flip whole cube, do A-perm-ish, flip back

Insert F2 from the right

A · B

A · B · A⁻¹

R F2 R'

set up, swap edges, undo

BLD edge cycle setup

A · B

A · B · A⁻¹

L R U' R' L'

a typical Beginner BLD edge insertion

Sledgehammer reposition

A · B

A · B · A⁻¹

U R' F R F' U'

shift the sledgehammer one U-turn over

Conjugation preserves order

Conjugation respects powers:

(ab a^{- 1})^{n} = a \cdot b^{n} \cdot a^{- 1}

. So b and

ab a^{- 1}

share the same order. On the cube: you can relocate any operation (a PLL, an F2L insertion, a commutator) to an equivalent location — its order and all internal properties are preserved.

8.2 Conjugacy classes

Elements that are conjugate to each other form a conjugacy class. All elements within a class share the same order, the same cycle type, and the same "topological action" — only the viewpoint differs. Two cube states in the same conjugacy class are essentially the same problem.

G has about 81,120 conjugacy classes (refined by Burnside's lemma under the cube's symmetry group). Each class is one "shape" of cube state. Rokicki's God's-number proof (§11) exploits exactly this structure to compress 4.3 × 10¹⁹ states to roughly 2 billion symmetry-equivalence classes.

Conjugate elements share the same cycle type. The table lists cycle types of a few common algs — note that superflip has identity perm on both corners and edges (it only flips EO without moving anything). Its "perm" cycle type is empty, the same as identity — they are distinguished only by orientation data.

Alg	Corner cycle type	Edge cycle type	Order	sgn
single face turn R	4-cycle	4-cycle	4	−1
sexy move R U R' U'	2-cycle × 2-cycle	3-cycle	6	+1
opposite-face pair R L	4-cycle × 4-cycle	4-cycle × 4-cycle	4	+1
Sune R U R' U R U2 R'	2-cycle × 2-cycle	3-cycle	6	+1
OLL 26 F R U' R' U' R U R' F'	4-cycle	4-cycle	4	−1
superflip U R2 F B R B2 R U2 L B2 R U' D' R2 F R' L B2 U2 F2	identity (no cycles)	identity (no cycles)	2	+1
checkerboard U2 D2 F2 B2 L2 R2	identity (no cycles)	2-cycle × 2-cycle × 2-cycle × 2-cycle × 2-cycle × 2-cycle	2	+1
anti-Sune R U2 R' U' R U' R'	2-cycle × 2-cycle	3-cycle	6	+1

The cube also has 48 outer symmetries (24 rotations × 2 mirror reflections). Burnside's lemma applied jointly with G gives the count of "truly distinct" states up to symmetry — see §18.

8.3 Conjugacy-class size — orbit–stabilizer

Definition 8.3 — centralizer

For

g \in G

, its centralizer

C_{G} (g) := {x \in G : xg = g x}

is the subgroup of elements that commute with g. It measures how much of G commutes with g.

Theorem 8.4 — orbit–stabilizer

The size of g's conjugacy class

[g] := {xg x^{- 1} : x \in G}

satisfies

∣ [g] ∣ = \frac{∣ G ∣}{∣ C _{G} ( g ) ∣}

i.e. "orbit size × stabilizer size = |G|". Hence

∣ [g] ∣

divides

∣ G ∣

This formula translates "how big is the conjugacy class" into "how many elements commute with g." Extremes:

Central elements (z ∈ Z(G)) commute with everything: $C_{G} (z) = G$ , so $∣ [z] ∣ = 1$ . Each owns a singleton class. For the cube, $Z (G) = {e, \textsc s u p er f l i p}$ (see §13), giving exactly two size-1 classes.
"Most non-trivial" elements: commute only with ⟨g⟩ itself, $∣ C_{G} (g) ∣ = ord (g)$ , so the class size is $∣ [g] ∣ = ∣ G ∣/ ord (g)$ . For order-1260 elements, $∣ [g] ∣ \leq 4.3 \times 1 0^{19} /1260 \approx 3.4 \times 1 0^{16}$ .

The class equation

Decomposing G into conjugacy classes gives

∣ G ∣ = ∣ Z (G) ∣ + [g] \neq \subset Z (G) \sum \frac{∣ G ∣}{∣ C _{G} ( g ) ∣}

where the first term counts central elements (size-1 classes) and the rest are larger classes. This is one of the deepest identities in finite group theory: it ties together the prime structure of |G|, the centre, and the non-trivial conjugation orbits in one line.

For the cube, $∣ Z (G) ∣ = 2$ and the total number of conjugacy classes is ≈ 81,120 (under joint Burnside with mirror symmetries), giving an average class size of $∣ G ∣/81, 120 \approx 5.3 \times 1 0^{14}$ . But the actual distribution is extremely uneven: a few huge classes (typical scrambles) account for almost all of |G|, while many small classes contain only hundreds of elements.

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