Janet Chen's Group Theory and the Rubik’s Cube PDF

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Moreover, a · 0 = 0 + a = 0 for all a ∈ ❘. 6. Often, we are interested in the case when the set A is the group itself. In this case, we say 32 that the group acts on itself. For instance, we can define a group action as follows: for g ∈ G and a ∈ G, define a · g = ag, the normal group multiplication of a and g (check that this defines a group action). We call this the action of G on itself by right multiplication. 7. If G acts on a set A, then the orbit of a ∈ A (under this action) is the set {a · g : g ∈ G}.

12. The image of φedge |ker ψcorner : ker ψcorner → S12 contains A12 . Proof. By [PS 9, #4], A12 is generated by the set of 3-cycles in A12 . 10, it suffices to show that every 3-cycle is in the image of φedge |ker ψcorner . 10, the strategy is to use conjugates of a single move to prove this. You shoudl have found a move in [PS 8, #9a] that does not affect any corner cubies but cycles 3 edge cubies. One such move is M0 = LR−1 U2 L−1 RB2 , which has disjoint cycle decomposition (ub uf db). Then, M0 ∈ ker ψcorner , and φedge (M0 ) = (ub uf db).

If a group action has only one orbit, we say that the action is transitive (or that the group acts transitively). 11. acts on the set of ordered pairs (C1 , C2 ) of different unoriented corner cubies. After all, if C1 and C2 are two different unoriented corner cubies, applying a move M ∈ sends these corner cubies to two different corner cubicles C1 and C2 . Then, we can define the group action by (C1 , C2 ) · M = (C1 , C2 ). ) By [PS 3, #5], this action is transitive. In the same way, ● ● acts on the set of ordered triples (C , C , C ) of different unoriented edge cubies.

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Group Theory and the Rubik’s Cube by Janet Chen

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