Abstract definitions for the symmetry groups of the regular polytopes, in terms of two generators. Part II: the rotation groups

Abstract

The complete (or “extended”) symmetry groups, investigated in Part I, are certain groups of orthogonal transformations, generated by reflections. Every such group has a subgroup of index two, consisting of those transformations which are of positive determinant (i.e., “movements” or “displacements”). The positive subgroup (in this sense) of [k1, k2, …, kn−1] is denoted by [k1, k2, …, kn−1]′, and is “the rotation group” (or, briefly, “the group”) of either of the regular polytopes {k1, k2, …, kn−1}, {kn−1, kn−2, …, k1}; e.g., [3, 4]′ is the octahedral group.