MAT301 ASSIGNMENT 3 DUE DATE: MONDAY JULY 6, 2015 AT THE BEGINNING OF LECTURE Question 1. Let x, y be elements of a group with |x| < , |y| < . If gcd(|x|, |y|) = 1, show that x y = {e}. Question 2. For any xed x G and H G, dene xHx1 = {xhx1 : h H}, and NG (H) = {g G : gHg 1 = H}. (a) Prove that xHx1 G for any x G. (b) If H is Abelian, prove that xHx1 is Abelian. (c) Prove that NG (H) G. (NG (H) is called the normalizer of H in G) Question 3. Suppose that H is a subgroup of Sn of odd order (n 2). Prove that H is a subgroup of An . Question 4. Show that in S7 , the equation x2 = (1 2 3 4) has no solutions but the equation x3 = (1 2 3 4) has at least two solutions. Question 5. Given that and are in S4 with = (1 4 3 2), = (1 2 4 3), and (1) = 4, determine and . Question 6. Let R = R {0}. Dene : GL(2, R) R via A det(A) (that is, (A) = det A). Note: R is a group under normal multiplication and GL(2, R) is a group under matrix multiplication. (a) Prove that is a group homomorphism. (b) Let SL(2, R) = {A GL(2, R) : det A = 1} Prove that ker = SL(2, R) (this should be a very short proof). Question 7. Suppose that : Z50 Z15 (both are groups under addition) is a group homomorphism with (7) = 6. (a) Determine (x) (you should give a formula for (x) in terms of x). (b) Determine the image of . (c) Determine the kernel of . 1