M417 Homework 9 Solutions Spring 2004
  1. (10 points) Let G be a group acting on a set S. Show that the orbits form a partition of S; i.e., S is the disjoint union of the orbits.

    Answer: Let x be in orbG(s) and in orbG(s'). Then x = gs and x = g's' for some g and g' in G. But this means that s = g-1g's'. Now let y be any element of orbG(s). Then y = hs for some h in G. Hence y = hs = hg-1g's' is in orbG(s'). Thus orbG(s') contains orbG(s). Similarly, orbG(s) contains orbG(s'). Thus any two orbits with a single element in common are actually the same subset of S. Since s is in orbG(s), every element of S is in some orbit. This means S is the union of the orbits, and the orbits are disjoint.

  2. Let G = { g1, ..., gm } be a finite group acting on a finite set S = { s1, ..., sn }. Let t be the number of distinct orbits. Let D = { (g, s) in GxS | gs = s }. Let p1 : D -> G be defined by p1((g, s)) = g, let p2 : D -> S be defined by p2((g, s)) = s, and let f : G -> Z be defined by taking f(g) = | { s in S | gs = s } |.
  3. (10 points) Let H : RxC -> C be defined as H((t, x + iy)) = (cos(t) + i sin(t))(x + iy). Show that this defines an action of the reals R (regarded as a group under addition) on the complex numbers C.

    Answer: We must show that h(t) defined by (h(t))(x + iy) = H((t, x + iy)) is a bijection on C, and that h is a homomorphism h from R to Bij(C). But h(t) is multiplication by cos(t) + i sin(t), and this is bijective since sin(t) + i cos(t) has the multiplicative inverse (cos(t) + i sin(t))-1 = cos(-t) + i sin(-t). And h(tt') = cos(tt') + i sin(tt') = (cos(t) + i sin(t))(cos(t') + i sin(t')) = h(t)h(t'), so h is a homomorphism.

  4. (10 points) For each complex number c, determine the orbit and stabilizer of c with respect to the action defined in the previous problem.

    Answer: If c = 0, then the orbit of c is just {0} (since 0 times any complex number is just 0), and the stabilizer of 0 is all of R. If c = x + iy is nonzero, write c as r(cos(s) + i sin(s)), where r = (x2 + y2)1/2 and s is some angle. Then the orbit of c is the circle of radius r (since (h(t))(x + iy) = r(cos(s+t) + i sin(s+t))), and the stabilizer of c is the subgroup of R comprising the integer multiples of 2Pi (since sine and cosine are periodic of period 2Pi).