Proving the order of quaternion group is 8

First, show $i^2=j^2$, then look at $i^4=ij^2i$

It may be useful to try to break the group representation down further. Canonically, the term $j^{-1}i$ is represented as an element unique from $i$ and $j$. Let's call this element $k$. What do you notice about the products $ij$, $jk$ and $ki$? Moreover, what are $i^2$, $j^2$ and $k^2$, and do each of these constructions look familiar? That may help you in showing why $i^4=1$.