Say, I'm given a group of quanterions $Q_{10}$ under multiplication whose set of elements $\left \{ i,j,k,x,y \right \}$ and I'm asked to generate the elements of the subgroup of $Q_{10}$, say, for the element x.
How I think this should be done is finding out what elements are generated by z: $\left \langle z \right \rangle=\left \{ z^{1},z^{2},\cdot \cdot \cdot \right \}$ up to where $z^{i}=z^{j}$ where the order of the element z is finite.
Am I right? Thanks in advance.