The quaternion group has the following presentation: \begin{eqnarray} Q=\langle i,j,k|ijk^{-1},ik^{-1}j^{-1},ij^{-1}k\rangle. \end{eqnarray} I have derived down $i^2=j^2=k^2$ by a proper multiplication of the words in the relation. My question is how to get $i^4=j^4=k^4=1$.
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1$ij=k,,jk=i \Rightarrow jkj=k$, but $ki=j$ so $kik^2i=k \Rightarrow k^2= i^{-2}$. – Derek Holt Jan 18 '20 at 09:10
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@DerekHolt Thanks for the nice brief answer! – Smart Yao Jan 18 '20 at 09:47