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.