Q 2. Given that jU22j = 10, determine the order of 7 in the group (U22; 22) by performing only three multiplications in (U22; 22). Explain your reasoning.

Policies of the University relating to Academic Integrity. (4) I may be subject to student discipline processes in the event of an act of academic misconduct. Each subsequent page must carry your name and your student number. Submissions that do not carry the handwritten Statement of Student Responsibility will not be marked. If you need an extension, you must arrange it with me before the due time. Questions Q 1. (a) Calculate the subgroup H:(12) of (A4;0). Write the elements in cycle notation. (b) Find the set of all left cosets of H in A4. [See PC7A, Q4 for a list of the elements of A4 ] Q 2. Given that U22=10, determine the order of 7 in the group (U22;22) by performing only three multiplications in (U22;22). Explain your reasoning. Q 3. Calculate 13257 in the group (U17;17). Q 4 . Let G be the group of symmetries (including flips) of the regular heptagon (7-gon) shown at right. As usual, we regard the elements of G as permutations of the set of vertex labels; thus, GS7. (a) I Let denote the rotation of the 7 -gon that takes the vertex 1 to the vertex 2. Write down the cyclic subgroup R:= as a set of elements of S7 in cycle notation. (b) What are the orders of each of the elements of R ? (c) Write down any symmetry G\R in cycle notation