MAT301 ASSIGNMENT 3

MAT301 ASSIGNMENT 3 = Question 1. Let x; y be elements of a group with jxj < 1; jyj < 1. If gcd(jxj; jyj) = 1, show that hxi \ hyi = feg. Question 2. For any xed x 2 G and H G, dene xHx??1 = fxhx??1 : h 2 Hg, and NG(H) = fg 2 G : gHg??1 = Hg. (a) Prove that xHx??1 G for any x 2 G. (b) If H is Abelian, prove that xHx??1 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 ?? f0g. Dene : GL(2;R) ! R via A 7! det(A) (that is, (A) = detA). 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) = fA 2 GL(2;R) : detA = 1g 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