Proposition. Local rotations of simplices and cubes [frap-000Z]
✍️source

    For \(d\geq 3\), a vertex figure of a \(d\)-simplex or a \(d\)-cube is a \((d-1)\)-simplex. Its automorphism group is \(S_d\), and its rotation subgroup is \(A_d\). These are the local corner groups used in [marchetti2025rubik, Section 4, p. 10, and Section 5, p. 14].

    Proof. 

      At a vertex of either polytope there are \(d\) incident edges. Faces containing that vertex correspond to subsets of those edges: for a simplex one chooses the other vertices, and for a cube one chooses the coordinate directions in which the face varies. This is the face poset of a simplex on \(d\) vertices, of dimension \(d-1\). An automorphism permutes its vertices, and every vertex permutation extends uniquely to the subsets, giving \(S_d\).

      A base flag orders these \(d\) vertices. Its distinguished involutions are adjacent transpositions. They generate \(S_d\), and the even words are precisely the even permutations. Thus the rotation subgroup is \(A_d\).

      Here \(d\) is abstract dimension. Varying the layer count of an \(N\times N\times N\) cube leaves the ambient dimension equal to three; it changes the piece orbits and allowed layer moves instead.