Exploring the G-Set Action of C on G=(R,+)

[hsong9]

Homework Statement

Let G = (R,+) and define az = e^iaz 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^iaz for z in C.
Let e^iaz = (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.
Correct??
I have no idea how to show and find orbits and stabilizers.
Thanks

 

[Dick]

Looks ok so far. You might want to elaborate a little on why a(bz)=e^(ia)(e^(ib)z)=(ab)z. If you think of the complex plane C as R^2, with coordinates (x,y), what kind of geometric operation does e^(ia)z correspond to?

 

[hsong9]

Thanks very much,
I think I can modify to more correct..
But I still have no idea about orbits and stabilizers.
Could you give me any hints?
Are the orbits just circle?
like, r is distance of z and denoted |z| and
e^ia is the circle which is 1 of radius??
Thanks!

 

[Dick]

Yes, the orbits are circles. e^(ia)z is z rotated around the origin by an angle a. Write z in polar form z=r*e^(it) to see this.

 

[hsong9]

and the stabilizers are when a = 0?

 

[Dick]

Depends on what value of z you are talking about. And the stabilizer of a given z is a subgroup. If you are saying it's the trivial subgroup {0} I don't agree with that.

 

[hsong9]

I do not know what you mean.
Yeah, I know the stabilizers are subgroup of G but
I don't know how I can apply to this problem..
I have to find in case of az = z, right?

 

[Dick]

Right. 1) suppose z=0, what's the stabilizer group? 2) suppose z is not zero. What about a=2pi?

 

[hsong9]

hmm,

I found something in my textbook,
If X is a G-set and x in X, the orbit of x is Gx = {ax | a in G}. combining this with "|Gx| = |G:S(x)| for each x in X" gives equivalent conditions that the orbit is a singleton:
Gx = {x} <=> ax=x for all a in G <=> S(x)=G.

And, when z = 0 and z is not zero, both cases seem just G = (R,+)..
Right??

 

[Dick]

If z=0 then a0=0 for all a. So, yes, the stabilizer is G. If z is not equal to zero that's definitely not true. Which rotation angles take a point to itself? It's not just zero. What does a rotation by 2pi do?!

 

[hsong9]

Thank you very much!