Quaternion Group

There's a motivation which helps setup the definition.

Consider the set:

G := {z \in C | z^{4} = 1}

Here $G = {1, - 1, i, - i} = ⟨ i ⟩$ . This is a Group under complex multiplication. Furthermore it's an Abelian Group.

We can generalize this process for various powers of $z$ . Namely the quaternions is generated from the set:

Q_{8} = {1, - 1, i, - i, j, - j, k, - k}

where the key relations are:

$i^{2} = j^{2} = k^{2} = - 1$ (notice that then they are all order 4 elements)
$i j k = - 1$ (notice that implies that $i = - k j$ , and other formulas)
To go in more detail, let's find more products that are formed from these relations.

i j

Let's find $i j$ :

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

j i

Let's find $j i$ :

\begin{aligned} i j k & = - 1 \\ i^{2} j k & = - i \\ - 1 j k & = - i \\ j k & = i \\ j^{2} k & = j i \\ - k & = j i \end{aligned}