Past exam of the short options course of the University of Oxford / 2015 / 2 / a / Solution

The defining relation $yx=x^{4}y$ lets us move every occurrence of $y$ to the right of every occurrence of $x$. More generally, repeated conjugation givesNegative powers cause no problem because $x^{-1}=x^8$ and $y^{-1}=y^2$. Thus every word in the group presentation can be put in the required form, reducing powers using $x^9=y^3=e$:At this stage existence is proved; uniqueness will follow from the concrete semidirect product in part (b). Moving $x^r$ past $y^j$ gives the multiplication lawThe action is consistent with reduction of $j$ modulo $3$ because $4^3\equiv 1(\mathrm{mod}9)$.