Group actions & Burnside — counting up to symmetry

So far we have treated G as the group itself — a set of elements with multiplication. But the real power of a group is in how it acts on other sets. G acts on the 26 cubies and their orientations; the 48-element outer symmetry group acts on G itself.

18.1 The 48 outer symmetries of the cube — group O_h

Beyond the internal cube group G, the cube as a 3D object has its own symmetry group of 48 elements — known in chemistry/crystallography as O_h. It decomposes into 10 conjugacy classes:

18.2 Natural cube actions

• G acts on the 54 stickers: each face turn sends stickers to new positions. The orbit decomposition: a 24-sticker orbit (corner stickers), a 24-sticker orbit (edge stickers), and 6 singleton orbits (centres).
• G acts on the 26 cubies: an 8-element orbit (corners), a 12-element orbit (edges), and 6 singleton orbits (centres). This tells us G "lives on" 8 corners + 12 edges.
• The 48-element outer symmetry group acts on G itself by conjugation. This is the action behind §8.2's conjugacy classes and §11's Rokicki proof.

18.3 Burnside's lemma (orbit counting)

Apply it to the 48-element outer cube symmetry group acting on G — that gives the count of "essentially distinct" cube states (ignoring whole-cube rotations and mirrors). For each symmetry g, Fix(g) is the set of cube states invariant under g.

Sum the fixed-point counts and divide by |D| = 48 (the order of the outer symmetry group):

This number is slightly bigger than |G| / 48 ≈ 9.01 × 10¹⁷ — because only a handful of states (like superflip) carry full 48-fold symmetry, while most states have none. So the "truly distinct" count is just a touch above the naive |G| / 48.

18.4 Orbit-stabiliser on the cube

Take X = 26 cubies (corners + edges), G acting. Pick any corner c (say URF, position 0). Its orbit G · c is all 8 corner positions, since G can send URF anywhere. Its stabiliser Stab(c) is the subgroup of operations fixing URF — order |G| / 8 = 5,406,500,409,311,232,000.

Pick an edge: |Orbit| = 12, |Stab| = |G| / 12 ≈ 3.6 × 10¹⁸. Orbit-stabiliser is the divide-and-conquer principle behind many cube solvers' two-table designs (one keyed by corners, one by edges).

18.5 Cayley's theorem — every group is a permutation group

The theorem is both obvious and stunning on the cube: G embeds in the symmetric group on |G| = 4.3 × 10¹⁹ elements. But in practice, G fits into S₈ × S₁₂ (dimension 8! + 12!) — the much tighter "natural embedding" that justifies why the (cp, ep) state vector suffices to describe a group element.

18.6 Conjugacy classes of Oₕ and typical Fix(g)

The 48-element $O_h$ acts on G; each $g\in O_h$'s fixed-point set $\mathrm{Fix}(g)$ depends on its conjugacy class. Typical values across the 10 classes:

A typical "k-fold symmetric" element gives $|\mathrm{Fix}(g)|\sim |G{|}^{1/k}$ (because k-fold symmetry pins each of cp, co, ep, eo onto its own k-orbit fixed locus). This is the practical numerical form of the Cauchy–Frobenius / Burnside count applied to the cube.

18.7 Pólya enumeration — 30 distinct 6-coloured cubes

The classic Pólya example: "how many essentially different ways to colour the 6 faces of a cube with 6 colours, modulo 24 rotations?" By Burnside:

For "exactly 6 distinct colours, one per face," the answer is $6!/24=30$. These are the 30 essentially distinct coloured cubes — but Erno Rubik (1974) used a single fixed colouring and let the stickers move around, giving 4.3 × 10¹⁹. Both counts are Burnside-style, separated by 18 orders of magnitude.