The Quaternion Group, Q8

# The Quaternion Group, Q8

Let $Q_8 = \{ 1, -1, i, \bar{i}, j, \bar{j}, k, \bar{k} \}$. The **Quaternion Group** is the set $Q_8$ with the operation $\cdot$ defined as follows:

$1$ | $-1$ | $i$ | $\bar{i}$ | $j$ | $\bar{j}$ | $k$ | $\bar{k}$ | |
---|---|---|---|---|---|---|---|---|

$1$ | $1$ | $-1$ | $i$ | $\bar{i}$ | $j$ | $\bar{j}$ | $k$ | $\bar{k}$ |

$-1$ | $-1$ | $1$ | $\bar{i}$ | $i$ | $\bar{j}$ | $j$ | $\bar{k}$ | $k$ |

$i$ | $i$ | $\bar{i}$ | $-1$ | $1$ | $k$ | $\bar{k}$ | $\bar{j}$ | $j$ |

$\bar{i}$ | $\bar{i}$ | $i$ | $1$ | $-1$ | $\bar{k}$ | $k$ | $j$ | $\bar{j}$ |

$j$ | $j$ | $\bar{j}$ | $\bar{k}$ | $k$ | $-1$ | $1$ | $i$ | $\bar{i}$ |

$\bar{j}$ | $\bar{j}$ | $j$ | $k$ | $\bar{k}$ | $1$ | $-1$ | $\bar{i}$ | $i$ |

$k$ | $k$ | $\bar{k}$ | $j$ | $\bar{j}$ | $\bar{i}$ | $i$ | $-1$ | $1$ |

$\bar{k}$ | $\bar{k}$ | $k$ | $\bar{j}$ | $j$ | $i$ | $\bar{i}$ | $1$ | $-1$ |

The quaternion group $Q_8$ is characterized by the equations:

(1)\begin{align} \quad (-1)^2 &= 1 \\ \quad i^2 &= j^2 = k^2 = -1 \\ \quad ijk &= -1 \end{align}

Observe that $ij = k$, $jk = i$, and $ki = j$. Meanwhile, $ji = \bar{k}$, $ik = \bar{j}$, and $kj = \bar{i}$. Therefore, $Q_8$ is a nonabelian group of order $8$.