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## Compute the order of each element in the quaternion group

Compute the order of each of the elements in $Q_8$.

Solution:

$x$            Reasoning            Order
1 1 is the identity. 1
-1 $(-1) \cdot (-1) = 1$ 2
$i$ $i \cdot i=-1$, $(-1) \cdot i = -i$, $(-i) \cdot i = 1$ 4
$-i$ $(-i) \cdot (-i)=-1$, $(-1) \cdot (-i) = i$, $i \cdot (-i) = 1$ 4
$j$ $j \cdot j = -1$, $(-1) \cdot j = -j$, $(-j) \cdot j = 1$ 4
$-j$ $(-j) \cdot (-j) = -1$, $(-1) \cdot (-j) = j$, $i \cdot (-j) = 1$ 4
$k$ $k \cdot k = -1$, $(-1) \cdot k = -k$, $(-k) \cdot k = 1$ 4
$-k$ $(-k) \cdot (-k) = -1$, $(-1) \cdot (-k) = k$, $k \cdot (-k) = 1$ 4
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