MAT301 ASSIGNMENT 4 DUE DATE: MONDAY JULY 20, 2015 AT THE BEGINNING OF LECTURE Question 1. (a) Prove that a group of order 8 must have an element of order 2. (b) Prove that a group of order 63 must have an element of order 3. Question 2. (a) Suppose : Z20 Z20 is an automorphism and (5) = 5. What are the possibilities for (x) (explain your answer)? (b) Suppose : Z3 Z5 Z15 is an isomorphism with (2, 3) = 2. Find the element in Z3 Z5 that maps to 1 (explain your answer). Question 3. Show that every automorphism of the rational numbers Q under addition to itself has the form (x) = x(1) (this is x times (1)). Question 4. Let be an automorphism of a group G. Prove that H = {x G : (x) = x} is a subgroup of G. Question 5. Suppose that G is a group with more than one element and G has no proper, non-trivial subgroups. Prove that |G| is prime. (Do not assume at the outset that G is nite, you must show this). Question 6. Let G be a nite group. (a) Let H and K be subgroups of G with H K G. Prove that [G : H] = [G : K][K : H]. (b) Prove that CG (a) = G if and only if a Z(G). (c) Use part (a) and (b) to prove that [G : Z(G)] cannot be a prime number. Question 7. (a)Determine Aut(Z). (b) Determine Aut(Z2 Z2 ). 1