1. Consider the symmetry group
D5 of the regular pentagon. The pentagon has vertices
1, 2, 3, 4, 5 in counterclockwise order, and the dihedral group has
r = (1 5 4 3 2) and l = (2 5)(3 4). Show that each of the five symmetries l, rl, r2l, r3l, r4l is simply a reflection of the pentagon about a line passing from a vertex to the midpoint of the opposite side.
2. (a) Let C be the set of complex numbers of norm 1 under multiplication. Recall that multiplication of complex numbers is defined as follows. If x = a + bi and y = c + di then xy = (ac - bd) + (ad + bc)i. The norm of a complex number a + bi is the distance of the point (a, b) to the origin (0, 0) in the plane (of complex numbers). In other words, the norm of x = a + bi is given by |x| = (a2 + b2)1/2. Show that C is a group under multiplication.
(b) Show that for every real number q, the complex number cos q + i sin q, which ought to be written as cos q + (sin q)i in strict complex number notation, is an element of the group C.
(c) Let R be the group of real numbers under addition. Show that the function F from R into C defined by F(r) = cos 2pr + i sin 2pr is a homomorphism of R onto C. (Don't be afraid to use some trig identities here.)
(d) What subgroup of
R is the kernel of F?
3. (Independent of problem 2.) Describe as best you can the symmetry group of a circle. (Hint: the circle behaves in many ways like a polygon with infinitely many sides. So use rotations and a fixed reflection like we did for n-gons.)