Lemma. the cyclic subgroup generated by the Hurwitz unit [fgap-000B]
✍️source

Let \[ \omega =\frac {-1+i+j+k}{2}. \] The calculation in a Hurwitz unit of order \(3\) gives \(\omega ^3=1\) and \(\omega \neq 1\). It follows that \[ C=\langle \omega \rangle =\{1,\omega ,\omega ^2\} \] is a cyclic subgroup of \(\mathbb {H}^{\times }\) of order 3. In particular, \(C\cong \mathbb {Z}/3\mathbb {Z}\). This subgroup appears in [voight2021quaternion, sec. 11.2.4, p. 168].

Proof.

Every power of \(\omega \) reduces modulo 3, so \(C\subseteq \{1,\omega ,\omega ^2\}\). The reverse inclusion is immediate. If \(\omega ^2=1\), multiplying by \(\omega \) would give \(\omega =1\), contrary to the coordinate formula. Thus the 3 displayed elements are distinct.

Lemma. the cyclic subgroup generated by the Hurwitz unit [fgap-000B]
✍️source

Proof.

Reference. Quaternion algebras [voight2021quaternion]

Quaternion and order-three subgroups [fgap-0007]

Definition. group action [fgap-0008]

Definition. the quaternion group inside Hamilton's Quaternions [fgap-0009]

Lemma. normal forms in the quaternion subgroup [fgap-000A]

Proof.

Lemma. the cyclic subgroup generated by the Hurwitz unit [fgap-000B]
✍️source

Proof.

Example. the Hurwitz action on the quaternion group [fgap-000C]

Remark. concrete and abstract factor models [fgap-000D]

Lemma. a Hurwitz unit of order \(3\) [fgap-0003]