UTM Fall 2015 MAT301 - Groups and Symmetries Problem Set 2 Due: At the beginning of your tutorial on Friday 25 September. 1. (3 marks) Give a specic example of some group G and elements g, h G where (gh)5 = g 5 h5 . 2. (5 marks) a) Let a = 4 and b = 11 be elements in the group Z18 . Compute a3 (ba)1 . b) Suppose G is a group with group operation 'addition modulo 12'. Suppose 10 G but 9 G. Find |G|. Carefully explain your reasoning. 3. (6 marks) Consider the group U (40). a) List the elements in U (40) and nd |U (40)|, i.e. the order of U (40). b) Find 71 (where 7 U (40)). c) Compute |13| (the order of the element 13 in U (40)). d) Find all x U (40) such that 13x = 7. (All computations in U (40).) Note: A brute force approach of simply trying all possibilities to see which ones work, will not give any marks. 4. (6 marks) Let g and h be elements in a group G. Prove that there exists an element x G such that xgx = h i there exists an element y G such that gh = y 2