Example. the twenty-four Hurwitz units [fgap-000K]
Example. the twenty-four Hurwitz units [fgap-000K]
The unique normal forms divide \(T\) into three disjoint rows: \[ \begin {array}{c|c|c} Q&Q\omega &Q\omega ^2\\ \hline 8\text { elements}&8\text { elements}&8\text { elements}. \end {array} \] Therefore \(|T|=8\cdot 3=24\).
The first row is \[ Q=\{\pm 1,\pm i,\pm j,\pm k\}. \] Since \[ \omega =\frac {-1+i+j+k}{2},\qquad \omega ^2=\frac {-1-i-j-k}{2}, \] left multiplication by the 8 elements of \(Q\) gives \[ Q\omega \sqcup Q\omega ^2 = \left \{ \frac {\epsilon _0+\epsilon _1i+\epsilon _2j+\epsilon _3k}{2}: \epsilon _r\in \{\pm 1\} \right \}. \] The coordinates are distinct, so these are the 16 half-integral units. Together with \(Q\), they are exactly the 24 Hurwitz units listed in [voight2021quaternion, sec. 11.2, p. 166]. Thus the subgroup \(T\) is the unit group of the Hurwitz order, not merely another group of the same cardinality.