M417 Homework 9 Spring 2004
Instructions: Solutions are due Fri., April 16.

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.
2. Let G = { g₁, ..., g_m } be a finite group acting on a finite set S = { s₁, ..., s_n }. 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 } |.
   • (10 points) Show that |stab_G(s₁)| + ... + |stab_G(s_n)| = t|G|. [Hint: Work with one orbit at a time, keeping in mind that the orbits partition S.]
   • (10 points) Show that f(g₁) + ... + f(g_m) = t|G|. [Hint: Show that f(g₁) + ... + f(g_m) = |D| = |stab_G(s₁)| + ... + |stab_G(s_n)|. Do this by showing that stab_G(s) = p₁(p₂^-1(s)), and f(g) = |p₁^-1(g)|. It may help to look at D in A₄x{1,2,3,4}, as in the example done in class of A₄ acting on {1,2,3,4}.]
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.
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.