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

Every element of \(Q\) has a unique normal form \[ i^a j^b,\qquad 0\leq a<4,\quad 0\leq b<2. \] Consequently, \[ Q=\langle i,j\rangle =\{i^a j^b:0\leq a<4,\ 0\leq b<2\} \] and \(Q\) has 8 elements.

Proof.

The relation \(ji=i^{-1}j\) moves each occurrence of \(j\) to the right. The relations \(i^4=1\) and \(j^2=i^2\) then reduce every word in \(i,j\) to the stated form. The 8 resulting quaternions are

\[ \begin {array}{cccc} 1&i&-1&-i\\ j&k&-j&-k. \end {array} \]

Their coordinates in \(\mathbb {H}=\mathbb {R}1\oplus \mathbb {R}i\oplus \mathbb {R}j\oplus \mathbb {R}k\) are distinct, so the normal forms are unique.

The two-row display will also be useful when the concrete subgroup is compared with an abstract presentation of \(Q_8\). See [voight2021quaternion, sec. 11.2, p. 166].

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

Proof.

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

Proof.

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

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