Algebra 1 Honours, ASE Rubik's Cube - Lecture 1 - Rubik's group and cycle notation

Oct 13, 2026

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Lecture 1 explains how the Rubik’s cube is modeled as a permutation group acting on cube states. It defines cubies, facelets, slots, and the state space S as bijections between facelets and slots, noting the count of possible states. The Rubik’s group G is defined via generators corresponding to 90-degree face rotations, and solving is framed as finding a group element mapping a target state back to the start state. Cycle notation for moves is constructed and illustrated, including how relabeling faces transforms commutators into related cycle decompositions.

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Lecture 1

Rubik's group: states, permutations, and generators

Moves, start state, and solving via group action

Cycle notation for cube moves

Relabeling faces and transforming commutators

Algebra 1 Honours, ASE Rubik's Cube We always compose and apply functions from left to right (opposite of the usual convention) . Lecture 1 We 􀀌rst give a careful de􀀌nition of the Rubik's group. De􀀌nition 1 . The cube is made up of 27 small cubes called cubies (there are three layers of nine cubies; one cubie is not visible, and in fact does not really exist) . Each cubie has one, two or three visible faces, called facelets. If a cubie has one facelet it is a center cubie, two facelets an edge cubie, and otherwise is a corner cubie. The positions occupied by cubies are called cubicles and the positions occupied by faclets are called slots. A state of the cube is 􀀘 a bijection ffaceletsg ! fslotsg. Note there are 54! states. We set S to be the set of states of the cube. Label the center cubies by F; B; U; D; L; R and the corresponding center cubicles by f; b; u; d; l; r (to label, we can pick F arbitrarily, and then choose U to be an arbitrary adjacent slot; the others are then determined) . We view moves of the cube as elements of Perm(S); and let F; B; U; D; L; R 2 Perm(S) be rotation by 90􀀎 clockwise around the given face (if one is viewing the face) . The Rubik's group G is the subgroup of Perm(S) they generate: G = hF; B; U; D; L; Ri 􀀒 Perm(S): We call elements of the Rubik's group moves. If 􀀛 2 S is a state and g 2 G is a move, the action of g on 􀀛, written 􀀛g, is the value of the permutation g on 􀀛 . We denote the start state (i.e., the state the cube arrives in) by e. The set P of possible states is P = feg j g 2 Gg: Solving the cube amounts to, given 􀀛 2 P ; 􀀌nd g 2 G such that 􀀛g = e: De􀀌nition 2 . We will label cubicles by the faces their facelets are on, e.g. , ur is the edge cubicle with one facelet on U and one facelet on R; the cubicle urf is the corner cubicle with one facelet on U, one facelet on R, and one facelet on F. We always label corner cubicles by listing the faclets in clockwise order, looking from the corner to the middle of the cube. So every corner can be written exactly three ways, e.g. , urf = rfu = fur, but not ruf: We can write elements of G using cycle notation, for instance, R = (urf; bru; drb; frd)(ur; br; dr; fr): This is telling us how R acts on any given state: the facelet in the u slot of the ufr cubicle is sent to the b slot of the bur cubicle, etc. In fact, when we write a cycle representing g 2 G, we are writing a bijection : fslotsg ! fslotsg such that 􀀛g = 􀀛 􀀎 = ffaceletsg 􀀀􀀛! fslotsg 􀀀! fslotsg; for all states 􀀛 2 S: Example 3 . To 􀀌nd the cycle notation of a move, pick a corner cubie and record the positions of its facelets under iterations of the move, until the cubie reaches the position it started in. Then pick a corner cubie that did not arise in the previous cycle, and repeat. Do this until all the corner cubies have been accounted for (if a cubie doesn't move, don't write anything) . Repeat this for edges, e.g. , RU = (urf; rfu; fur)(ufl; ulb; ubr; bdr; dfr; luf; bul; rub; rbd; rdf; flu; lbu; bru; drb; frd)(fr; uf; ul; ub; ur; br; dr): This example shows that the order we write the adjacent facelets of a cubicle is important:(urf; rfu; fur) is not (urf; urf; urf) because the later is the identity, but RU does move the facelets currently in the urf cubicle: it rotates them counterclockwise. Similarly, (RU)5 2 rotates the facelets in the ubr cubicle counterclockwise. We can shorten cycle notation by incorporating these twists, e.g. , RU = (urf )+ (ufl; ulb; ubr; bdr; dfr) 􀀀 (fr; uf; ul; ub; ur; br; dr) URU 􀀀1R 􀀀1 = (ulb; ubr)+ (dfr; rfu) 􀀀 (ur; fr; ub): Here the + means clockwise, and 􀀀 means counterclockwise. We can do the same for edges, e.g. , FRUR 􀀀1U 􀀀1F 􀀀1U = (uf )+ (ul; bu)+ (ufl)+ (ulb; fur) 􀀀 : Example 4 . Finding cycle notation is a bit tedious. We can use relabeling of the faces to help. For instance, we found the cycle decomposition of [U; R] = URU 􀀀1R 􀀀1 above; can we relabel to 􀀌nd the cycle decomposition of [U;

FAQs

How is a Rubik's cube state defined in Lecture 1?

A state is defined as a bijection from the set of facelets to the set of slots. The cube’s state space P is the set of states reachable from the start state by group moves.

What is the Rubik's group G in this lecture?

G is the subgroup of permutations of the state space generated by quarter-turn rotations of the six faces F, B, U, D, L, and R. Elements of G are called moves.

How is cycle notation used to describe a move?

To write a move’s cycle notation, track how a selected cubie’s facelets move under repeated application of the move until returning to the start position. Repeat for all cubies, omitting those that do not move, and use clockwise/counterclockwise twist conventions where needed.

Oct 13, 2026

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