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    $-1$ is the unique element of order $2$ in the quaternion group

    E-AMD-DI5UYFIZ

    Area
    Algebra
    Topics
    Group Presentations, Groups
    Status
    draft

    Show that the Quaternion group has only one element of order 2, namely \(-1\).

    Solution

    The quaternion group is \[ Q_8=\{\pm1,\pm i,\pm j,\pm k\}, \] with \[ i^2=j^2=k^2=-1. \] Now \[ (-1)^2=1, \] and \(-1\ne1\), so \(-1\) has order \(2\).

    For each \(u\in\{i,j,k\}\), \[ (\pm u)^2=u^2=-1\ne1, \qquad (\pm u)^4=1. \] Hence each of \(\pm i,\pm j,\pm k\) has order \(4\). The identity has order \(1\).

    Therefore the only element of order \(2\) is \[ \boxed{-1}. \]

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