M417 Homework 9 Spring 2004

*Instructions*: Solutions are due Fri., April 16.

- (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.
- Let G = { g
_{1}, ..., g_{m} }
be a finite group acting on a finite set S =
{ s_{1}, ..., s_{n} }.
Let t be the number of distinct orbits.
Let D = { (g, s) in GxS | gs = s }. Let p_{1} : D -> G
be defined by p_{1}((g, s)) = g, let
p_{2} : D -> S be defined by p_{2}((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}(s_{1})| + ... + |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(g
_{1}) + ... + f(g_{m}) =
t|G|. [Hint: Show that f(g_{1}) + ... + f(g_{m}) = |D| =
|stab_{G}(s_{1})| + ... + |stab_{G}(s_{n})|.
Do this by showing that stab_{G}(s) = p_{1}(p_{2}^{-1}(s)), and
f(g) = |p_{1}^{-1}(g)|. It may help to look
at D in A_{4}x{1,2,3,4}, as in the example
done in class of A_{4} acting on {1,2,3,4}.]

- (10 points) Let H :
**R**x**C** -> **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**.
- (10 points) For each complex number c, determine the orbit and stabilizer
of c with respect to the action defined in the previous problem.