What is the quaternion group \(Q_8\)? Give its presentation and list the number of elements of each order.
Solution
The quaternion group is \[ Q_8=\{\pm1,\pm i,\pm j,\pm k\}, \] with multiplication determined by \[ i^2=j^2=k^2=ijk=-1. \] Equivalently, \[ Q_8=\langle i,j\mid i^4=1,\ i^2=j^2,\ jij^{-1}=i^{-1}\rangle. \]
The element orders are immediate from the relations: \[ \operatorname{ord}(1)=1,\qquad \operatorname{ord}(-1)=2, \] and each of \(\pm i,\pm j,\pm k\) squares to \(-1\), so each has order \(4\). Thus the counts are \[ \begin{array}{c|ccc} \text{order}&1&2&4\\\hline \text{number of elements}&1&1&6. \end{array} \] In particular, \(-1\) is the unique involution of \(Q_8\).