Example. conjugation by a Hurwitz unit [fgap-0004]
Example. conjugation by a Hurwitz unit [fgap-0004]
Consider left conjugation by the Hurwitz unit \(\omega =(-1+i+j+k)/2\). We have \[ \omega i\omega ^{-1}=k,\qquad \omega j\omega ^{-1}=i,\qquad \omega k\omega ^{-1}=j. \] Thus the cycle is \(i\mapsto k\mapsto j\mapsto i\). Voight says that conjugation cyclically rotates these units in [voight2021quaternion, sec. 11.2.4, p. 168]; the calculation here fixes the direction. Wilson discusses an abstract order-3 automorphism cycling the quaternion generators in [wilson2021finite, sec. 4.3, pp. 12--13, v5]. That discussion does not select the exact quaternion or fix the oriented cycle calculated here.
It is enough to calculate \[ \omega i=k\omega ,\qquad \omega j=i\omega ,\qquad \omega k=j\omega . \] For example, \[ \omega i =\frac {-1-i+j-k}{2} =k\omega . \] Right multiplication by \(\omega ^{-1}\) proves the three conjugation equations.