2 Math 113 Summer 2016. Due Thursday June 30th Make sure to write your solutions to the following problems in complete English sentences. Solutions that are unreadable or incoherent will receive no credit. Provide complete justifications for all claims that you make. Problems will be of varying difficulty, and do not appear in any order of difficulty. 1. Let g be an element of order k in a group G . a) If f : G H is a homomorphism, prove that the order of f (g ) divides k. b) If f : G H is an isomorphism, prove that the order of f (g ) is equal to k. 2. Find all automorphisms of Z/4Z. 3. Does k x = x + k define an action of Z/nZ on R? 4. Let U3 (Z/2Z) be the group of matrices 1 a b def U3 (Z/2Z) = 0 1 c | a, b, c Z/2Z 0 0 1 with matrix multiplication as the binary operation. a) Show that |U3 (Z/2Z)| = 8. b) Find an element R of order 4 and an element S of order 2 in U3 (Z/2Z), such that SRS = R 1 . c) Write the group D8 as {e, r , r 2 , r 3 , s, sr , sr 2 , sr 3 }, where r is counterclockwise 90 rotation of the plane and s is a reflection. Consider the map f : D8 U3 (Z/2Z) which sends s i r j to S i R j . This defines a homomorphism (you do not need to prove this). Prove that f is actually an isomorphism. 5. (Important, and useful for the following exercises!) Prove that \"a homomorphism is determined by what it does to the generators\