Automorphisms and Subrings of O

This chapter examines the octavian units in some detail, and then uses them to find the automorphism group of O = O⁸ and to study some of its subrings. It presents a little table that gives the quaternion triplets abc (for which i_a, i_b, i_c behave like i, j, k) in its first column, which can be supplemented by ∞ or 0 to give seven “quads” that are halving-sets for the octavians, as are their complements. The chapter examines the automorphism group Aut(O) of the ring O octavian integers. We can obtain an automorphism permuting the vertices of any triangle in the hyperhexagon, and since the graph is connected, these establish the transitivity. The chapter briefly discusses the transitivities in the nontrivial cases, paying particular attention to those that correspond to maximal subgroups. The transitivity on rings of type G² and G⁸ follows since these are in 1–1 correspondence with each other and with the vertices of the hyperhexagon.