(i) Write p as a product of disjoint cycles. (ii) Find pq and qp.(iii) Write p as a product of the transpositions (12), (13), (14), (15), (16). (iv) Determine if q is an even or odd permutation.(v) Write q as a product of 3-cycles.
Let G be a group.(i) Let w 2 G. Prove that the inverse of w is unique. (ii) Let x, y 2 G. Prove that (xy)1 = y1x1.
For each of the following sets, either prove that it forms a group under matrix multiplication, or explain why it is not a group.