Mathematics 581   Problem Set 6   Due Monday, May 20

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 two generators
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 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  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.)