Proving g-Orbit of z is Invariant Under g

[zcdfhn]

Suppose g$$\in$$ Isom C, z$$\in$$ C:

Prove that the g-orbit of z is invariant under g.

I just need some clarification on what this is asking for:

1.) Are we assuming that g is a group of the isometries of C under composition?
2.) To show invariance, would I only have to show that the g-orbit of z $$\in$$ C?


Here's my guess at the proof:

The g-orbit of z is defined as Orbit_g(x) = {h(x)|h$$\in$$ g}, x$$\in$$ C.
Now, for all h$$\in$$ g, h(z)$$\in$$ C since h$$\in$$ Isom C, which means h is an isometry from the complex plane to itself. QED.

Thanks in advance.

 

[Dick]

What does it mean for h to be an element of g if g is an element of Isom(C)? g isn't a set, is it? Can you clarify your definitions?