Definition. The rotation subgroup [marchetti2025rubik, Definition 4] [frap-000K]
Definition. The rotation subgroup [marchetti2025rubik, Definition 4] [frap-000K]
Let \(Q\) be a regular abstract polytope of rank \(m\geq 2\). Choose a base flag and its distinguished generators (involutions) \(\rho _0,\ldots ,\rho _{m-1}\). Its rotation subgroup is
\[\Gamma ^+(Q)=\langle \rho _{j-1}\rho _j:1\leq j<m\rangle .\]The index starts at \(1\): for a polygon, the single adjacent pair \(\rho _0\rho _1\) generates its rotations. For triangles and squares these are the usual cyclic rotation groups.
Equivalently, \(\Gamma ^+(Q)\) consists of the elements represented by even-length words in the distinguished involutions. Indeed every pair \(\rho _i\rho _j\) is a product of adjacent pairs or their inverses, by cancellation along the index interval. Conjugating an even word by a distinguished involution again gives an even word, so this subgroup is normal. A change of base flag conjugates the distinguished generators and therefore leaves the subgroup unchanged. We do not require it to have index two for every abstract polytope.