§2

The cube group G

Write the solved state as $e$ . Each "turn a face 90° or 180°" is a permutation acting on the positions and orientations of the 26 cubies. The set G consists of all permutations producible by composing such turns.

Definition 2.1 — the cube group

G = ⟨ U, D, L, R, F, B ⟩

The group generated by the six face turns U, D, L, R, F, B. Each generator is a 90° clockwise face turn; its inverse is the CCW turn (U′), its square is the half-turn (U2), giving the standard working set of 18 moves.

The six generators — click any face below; cubing.js will animate it.

Down

Right

Left

Front

Back

Aside: the slice moves M, E, S and full rotations x, y, z are not extra generators of G. They are derived: M = R' L x', etc. They exist only as a notational convenience.

2.1 Generating set — 18 or 6?

The minimal generating set has 6 elements (the six face turns), but in notation we usually write 18: each face × 3 angles (90°, 180°, 270° = the inverse U′). These 18 constitute the Half-Turn Metric (HTM) used in WCA competition.

The stricter Quarter-Turn Metric (QTM) only allows 90° turns (180° counts as two). In QTM, the generating set has 12 elements, and the group's diameter rises to 26.

Definition 2.2 — metric on a group

Given a generating set

S

, for any

g \in G

, define

∣ g ∣_{S}

as the minimum number of S-tokens whose product equals g. HTM uses the 18-generator set; QTM the 12-generator set. The "optimal solution length" is simply

∣ g ∣_{S}

2.2 Group presentation

A group can also be presented abstractly as "generators with relations." For instance $Z /4 = ⟨ a ∣ a^{4} = e ⟩$ means "one element a, which when raised to the 4th power gives e." For the cube:

G = ⟨ U, D, L, R, F, B U^{4} = D^{4} = L^{4} = R^{4} = F^{4} = B^{4} = e, U D = D U, L R = R L, F B = B F, (plus dozens of opaque longer relators) ⟩

Beyond the trivial "each face cycles in 4" and the "parallel faces commute" relations, the rest of the relators are too tangled to enumerate cleanly. A complete finite presentation for G is mathematically curious but rarely written down explicitly.

Curiously, the word problem (decide if two words represent the same element) is solvable for G — because G is finite, normalising via the cube state suffices. But the shortest-word problem (find the optimal representation of g) is NP-hard in the general setting. That's why optimal solvers are genuinely difficult.

2.3 Subgroups

Any subset of generators generates a subgroup — usually much smaller than G. Common ones:

$⟨ U ⟩$ — only U-turns allowed: isomorphic to $Z /4$ (4 elements)
$⟨ U, D ⟩$ — up & down faces: isomorphic to $Z /4 \times Z /4$ = 16 elements
$⟨ R, U ⟩$ — The subgroup generated by R and U: order ≈ 73 million. A classic Cayley-graph subject.
$⟨ U^{2}, D^{2}, L^{2}, R^{2}, F^{2}, B^{2} ⟩ = G_{3}$ — half-turns only, 663,552 elements (the "domino group")

2.4 The 18 face turns — complete list

Face	90°	180°	270° = 90° CCW	HTM count
U	U	U2	U'	3
D	D	D2	D'	3
L	L	L2	L'	3
R	R	R2	R'	3
F	F	F2	F'	3
B	B	B2	B'	3
Total				18

2.5 Cycle structure of each face turn

Each face turn is a permutation of the 8 corners and 12 edges. Writing R out explicitly (using the cubie indexing of §3):

R = (0374)_{c} \cdot (01148)_{e}

That is the 4-cycle URF → UBR → DRB → DFR → URF on corners, plus the matching 4-cycle UR → BR → DR → FR → UR on edges. Likewise U is $(0123)_{c} (0123)_{e}$ , and F also flips 4 edges (EO+1). All six generators follow the same pattern: 4-cycle on corners × 4-cycle on edges + optional orientation kick.

Gen	Corner cycle	Edge cycle	Order	Δ CO	Δ EO
U	$(0123)$	$(0123)$	4	0	0
R	$(0374)$	$(01148)$	4	yes	0
F	$(0451)$	$(1859)$	4	yes	4 edges +1

2.6 Relations and a non-relation

Obvious relations: each face has order 4, $U^{4} = e$ , and trivially $U \cdot U^{'} = e$ . But U does not commute with R:

U R \neq = R U (verify: apply both to the solved state; resulting corners and edges differ)

That non-relation is the entire source of cube theory — if UR = RU the cube would collapse to Abelian. A Coxeter-style presentation would demand $(s_{i} s_{j})^{m_{ij}} = e$ for some small $m_{ij}$ . G is not a Coxeter group: there is no small integer k with $(U R)^{k} = e$ ( $∣ U R ∣ = 105$ , see §8). G is genuinely "wilder" than the standard symmetry groups.

∎

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