- #1

- 80

- 1

## Homework Statement

Let G = (R,+) and define az = e

^{ia}z for all z in C and a in G. Show that this definition makes C into a G-set, describe the action geometrically, and find the orbits and the stabilizers.

## The Attempt at a Solution

A mapping G x C -> C, denoted (a,z) -> az = e

^{ia}z for z in C.

Let e

^{ia}z = (cos(a) +

*i*sin(a))(x+

*i*y)

Thus az = a(x+

*i*y) = (cos(a) +

*i*sin(a))(x+

*i*y)

We can be called this an action of G as satisfied the follwing:

Since 0 is identity of G, 0z = (cos(0) +

*i*sin(0))(x+

*i*y) = (x+

*i*y) = z

Also this satisfies such as a(bz) = (ab)z for all z in C and for all a,b in G

Thus G acts on C, called a G-set.

Correct??

I have no idea how to show and find orbits and stabilizers.

Thanks