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 $8$.