Mathematics

Mathematics

Question 1. Let x, y be elements of a group with |x| < 8, |y| < 8. If gcd(|x|, |y|) = 1, show that

hxi n hyi = {e}.

Question 2. For any fixed x ? G and H  G, define xHx-1 = {xhx-1

: h ? H}, and

NG(H) = {g ? G : gHg-1 = H}.

(a) Prove that xHx-1  G for any x ? 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 x

2 = (1 2 3 4) has no solutions but the equation x

3 = (1 2 3 4)

has at least two solutions.

Question 5. Given that ß and ? are in S4 with ß? = (1 4 3 2), ?ß = (1 2 4 3), and ß(1) = 4, determine ß

and ?.

Question 6. Let R

× = R – {0}. Define f : GL(2, R) ? R

× via A 7? det(A) (that is, f(A) = det A). Note:

R

× is a group under normal multiplication and GL(2, R) is a group under matrix multiplication.

(a) Prove that f is a group homomorphism.

(b) Let

SL(2, R) = {A ? GL(2, R) : det A = 1}

Prove that ker f = SL(2, R) (this should be a very short proof).

Question 7. Suppose that f : Z50 ? Z15 (both are groups under addition) is a group homomorphism with

f(7) = 6.

(a) Determine f(x) (you should give a formula for f(x) in terms of x).

(b) Determine the image of f.

(c) Determine the kernel of f.