Let G = (R,+) and define az = eiaz 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 = eiaz for z in C.
Let eiaz = (cos(a) + isin(a))(x+iy)
Thus az = a(x+iy) = (cos(a) + isin(a))(x+iy)
We can be called this an action of G as satisfied the follwing:
Since 0 is identity of G, 0z = (cos(0) + isin(0))(x+iy) = (x+iy) = 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.
I have no idea how to show and find orbits and stabilizers.