Example. the binary tetrahedral subgroup of the quaternions [fgap-000J]
Example. the binary tetrahedral subgroup of the quaternions [fgap-000J]
Inside \(\mathbb {H}^{\times }\), let \[ T=Q\vee C. \] The Hurwitz action in the Hurwitz action on the quaternion group says that \(C\) normalizes \(Q\). Hence a normalized product is the generated subgroup gives \[ T=QC,\qquad Q\mathrel {\trianglelefteq }T. \]
The groups \(Q\) and \(C\) have orders 8 and 3. The order of their intersection divides both, so \[ Q\cap C=\{1\}. \] They are therefore complementary in \(T\). Every element has a unique normal form \[ q\omega ^r,\qquad q\in Q,\quad 0\leq r<3, \] and the internal semidirect-product theorem gives \[ Q\rtimes _{\alpha }C\cong T,\qquad \alpha (c)(q)=cqc^{-1}. \]
The action orientation is fixed by the quaternion calculation: \[ i\longmapsto k\longmapsto j\longmapsto i. \] Voight identifies this semidirect product with the Hurwitz unit group and the binary tetrahedral group in [voight2021quaternion, sec. 11.2.4, p. 168].