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 $8$.