1 Ryerson University - MTH 617 Assignment 3 Date due: Wednesday, March 8, 2017 Question 1 Suppose that G is a group. (i) If for all a, b and c in G, ab = ca implies c = b, show that G is abelian. (ii) If a, b are distinct elements in G, show that either a2017 6= b2017 or a2018 6= b2018 . Question 2 (i) Let H < Z with H 6= Z. If p H and p is a prime, give as much information about H as you can. (ii) G be a group. For each a G, define C(a) := {x G : xa = ax}. Show that C(a) < G and C(a) = C(a1 ). Question 3 Read Chapter 10 and solve #C6, #D4, #D6, #E1, #F3 of Chapter 10. Question 4 (i) Suppose that a is an integer such that gcd(a, 32760) = 1. Compute [a12 ]32760 . 1. Is it true that 2017 617 and 1717 have the same remainder when divided by 100? Justify your