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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 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 .

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