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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$$i \cdot i=-1$, $(-1) \cdot i = -i$, $(-i) \cdot i = 1$4
$-i$$(-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


Linearity

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