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THEORY · GROUP THEORY

The Rubik's Cube,
as a Group

43 quintillion positions is not chaos. It is a beautifully structured algebraic object. An illustrated, interactive primer.

cuberoot · 2026 · 32 sections · 40+ interactive & visual panels · KaTeX-rendered math
4.33 × 1019
|G|
reachable cube states
20
God's number (HTM)
group diameter — longest optimal solve
31 + 45
sections · interactive panels
KaTeX formulas · cubing.js animations
core theorem · closed form for |G|
∣G∣=28!⋅37⋅12!⋅211​=43,252,003,274,489,856,000
8!corner perms
37corner twists
Σco ≡ 0
12!edge perms
211edge flips
Σeo ≡ 0
÷ 2parity match
sgn(c) = sgn(e)
= 227 · 314 · 53 · 72 · 11→ §4 full derivation→ §5 why ÷ 2 / ÷ 3 / ÷ 2
feature · SUPERFLIP

All edges flipped — one of three positions maximally far from solved

Every edge sits in its home slot, but all are flipped (cp​=e, ep​=e, co​=0, eo​=(1,1,…,1)). Solvable in exactly 20 HTM moves, and no fewer — the lower bound nailed down first when Rokicki et al. proved God's number = 20 in 2010.

U R2 F B R B2 R U2 L B2 R U' D' R2 F R' L B2 U2 F2
§11 God's number = 20 ↗§7 order of an element ↗§13 pattern gallery ↗
highlights · four pivotal ideas
§1
Four axioms
G1​G2​G3​G4​
Closure · associativity · identity · inverse — why the cube literally is a group
→
§5
Three invariants + proofs
∑co​≡0,∑eo​≡0
Σco mod 3, Σeo mod 2, parity match — why only 1/12 of "free" states are reachable
→
§11
God's number = 20
diam(Γ(G,S))=20
35 CPU-years brute-forced 4.3 × 10¹⁹ states — none needs 21 moves
→
§14
Cayley graph
Γ(G,S)
Vertices = states · edges = face turns · diameter = God's number · BFS = optimal solver
→
contents · 32 sections, grouped by theme
§1 – §6Foundationsaxioms · generators · state vector · order · invariants · structure theorem
  • §1What is a group?
  • §2The cube group G
  • §3State vector
  • §4The order |G|
  • §5Three invariants + proofs
  • §6Structure theorem
§7 – §11Core group theoryelement order · conjugation · commutators · Thistlethwaite chain · God's number
  • §7Order of an element
  • §8Conjugation
  • §9Commutators + centre
  • §10Subgroup chain
  • §11God's number = 20
§12 – §16Extensions · geometry & patternsbeyond · pattern gallery · Cayley graph · other puzzles · open problems
  • §12Beyond the cube
  • §13Famous patterns
  • §14Cayley graph
  • §15Other twisting puzzles
  • §16Open problems
§17 – §21Advanced algebrahomomorphisms · actions + Burnside · Lagrange + cosets · quotients · S_n / A_n
  • §17Homomorphisms
  • §18Group actions + Burnside
  • §19Lagrange + cosets
  • §20Normal subgroups + quotients
  • §21Symmetric & alternating groups
§22 – §26Computation · algorithms · representationsolving algorithms · distance distribution · random walks · BSGS · representation theory
  • §22Solving algorithms
  • §23Distance distribution
  • §24Random walks on G
  • §25Computational group theory
  • §26A glimpse of representation theory
§27 – §32Puzzle mathematics · jaapsch.netLights Out · peg solitaire · Hamilton · PGL₂(𝔽₅) · rotational graph puzzles · useful math
  • §27Lights Out · linear algebra over 𝔽₂
  • §28Peg solitaire · 3-colouring invariant
  • §29Hamiltonian paths + Gray codes
  • §30Two-face corners ≅ PGL₂(𝔽₅) ≅ S₅
  • §31Rotational puzzles on graphs · (x,y,z)
  • §32Useful mathematics · permutation visualiser
REFReferences12 entries · textbooks · papers · web resources
  • REFBibliography
∎
cuberoot.me · Rubik's Cube as a Group · 2026