Question

\[ x^{13}=y^{3}=1, y x=x^{3} y . \] Every element of \( G \) can be written uniquely as \( x^{i} y^{j} \) where \( 0 \leqslant i \leqslant 12 \) and \( 0 \leqslant j \leqslant 2 \). (You do not need to prove this fact.) (a) Show that yx^i=x^3i y for all integersi. (b) Prove that the centraliser CG​ (y) is {y^j:0⩽j⩽2}. Use this to find the size of the conjugacy classClG​(y). (c) Prove thatClG​(y)={x^iy:0⩽i⩽12}