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From Wikipedia:

In group theory, the quaternion group $Q_8$ (sometimes just denoted by $Q$) is a non-abelian group of order eight, isomorphic to a certain eight-element subset of the quaternions under multiplication.

There are many representation of $Q_8$, in one dimension, in two dimension etc. enter image description here

Question : How can I understand this group? Please explain simply. I have read its definition on Wikipedia but I did not get more than that it is non-abelian group of order eight. I am looking for simplest representation of it. What is the underlying operation? How many elements of order two, four, eight are there?

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    $\begingroup$ You should be able to see for yourself that it could not possibly have an element of order $8$. It has a unique element of order $2$. $\endgroup$ – Derek Holt Feb 13 '18 at 14:10
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$$\begin{array}{cccc} 1 = \left(\!\!\begin{array}{rr}1 & 0\\0&1\end{array}\!\!\right),& x = \left(\!\!\begin{array}{rr}0 & 1\\-1 & 0\end{array}\!\!\right),& x^2 = \left(\!\!\begin{array}{rr}-1 & 0\\0 & -1\end{array}\!\!\right),& x^3 = \left(\!\!\begin{array}{rr}0 & -1\\1 & 0\end{array}\!\!\right),\\ \\ y = \left(\!\!\begin{array}{rr}i & 0\\0 & -i\end{array}\!\!\right),& xy = \left(\!\!\begin{array}{rr}0 & -i\\-i & 0\end{array}\!\!\right),& x^2y = \left(\!\!\begin{array}{rr}-i & 0\\0 & i\end{array}\!\!\right),& x^3y = \left(\!\!\begin{array}{rr}0 & i\\i & 0\end{array}\!\!\right). \end{array}$$

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