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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
11 is the identity.1
-1$(-1) \cdot (-1) = 1$2
$i$$i \cdot i=-1, (-1) \cdot i = -i, (-i) \cdot i = 14 -i$$(-i) \cdot (-i)=-1$, $(-1) \cdot (-i) = i$, $i \cdot (-i) = 1$4
$j$$j \cdot j = -1, (-1) \cdot j = -j, (-j) \cdot j = 14 -j$$(-j) \cdot (-j) = -1$, $(-1) \cdot (-j) = j$, $i \cdot (-j) = 1$4
$k$$k \cdot k = -1, (-1) \cdot k = -k, (-k) \cdot k = 14 -k$$(-k) \cdot (-k) = -1$, $(-1) \cdot (-k) = k$, $k \cdot (-k) = 1$4