Math 561 H
Homework 5
Fall 2011
Drew Armstrong
1. We saw in class that any element of the orthogonal group O(2) has the form
R :=
cos sin
sin cos
or
F :=
cos sin
.
sin cos
The matrix R (with determinant 1) rotates the plane around 0 counterclockwise b
Math 561 H
Homework 4
Fall 2011
Drew Armstrong
1. Dene the ring of quaternions H := cfw_a1 + bi + cj + dk : a, b, c, d R, with the relations
1 = 1 and i2 = j2 = k2 = ijk = 1. Dene the quaternion absolute value by
|a1 + bi + cj + dk|2 := a2 + b2 + c2 + d2
Math 561 H
Homework 3
Fall 2011
Drew Armstrong
Group Problems.
1. Let G be a group. Given a G dene the centralizer Z(a) := cfw_b G : ab = ba. Prove that
Z(a) G. For which a G is Z(a) = G?
2. We say a, b G are conjugate if there exists g G such that a = gb
Math 561 H
Homework 2
Fall 2011
Drew Armstrong
1. Let : G H be a homomorphism of groups. Prove that im is a subgroup of H.
2. Let G be a set with binary operation (a, b) ab and consider the following possible axioms:
(1) a, b G, a(bc) = (ab)c.
(2) e G, a