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
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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