Definition. The Rubik group and labelled configurations [frap-000N]
Definition. The Rubik group and labelled configurations [frap-000N]
For a finite regular abstract polytope \(P\) of rank \(n\geq 3\), the Rubik group is the subgroup generated by all facet moves:
\[\operatorname {Rub}(P)= \langle \mu _{H,\alpha }:H\text { a facet},\quad \alpha \in \Gamma ^+(H/F_{-1})\rangle \leq \operatorname {Sym}(\mathcal R(P)).\]This is Marchetti's generated group of permutations [marchetti2025rubik, Section 3, p. 6, following Definition 6]. The inverse lemma ensures that the generators are permutations.
Give each incidence position its own initial label. A labelled configuration can be represented by the permutation carrying each initial label to its current position. The identity is the solved configuration; the reachable labelled configurations are the elements of \(\operatorname {Rub}(P)\). A new move acts on the left: applying \(\mu \) to configuration \(g\) gives \(\mu g\), with \(g\) acting first. A visible coloring may identify several labelled configurations, so a solved visible coloring need not determine the labelled permutation.