- #1
zcdfhn
- 23
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Suppose g[tex]\in[/tex] Isom C, z[tex]\in[/tex] 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 [tex]\in[/tex] C?
Here's my guess at the proof:
The g-orbit of z is defined as Orbitg(x) = {h(x)|h[tex]\in[/tex] g}, x[tex]\in[/tex] C.
Now, for all h[tex]\in[/tex] g, h(z)[tex]\in[/tex] C since h[tex]\in[/tex] Isom C, which means h is an isometry from the complex plane to itself. QED.
Thanks in advance.
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 [tex]\in[/tex] C?
Here's my guess at the proof:
The g-orbit of z is defined as Orbitg(x) = {h(x)|h[tex]\in[/tex] g}, x[tex]\in[/tex] C.
Now, for all h[tex]\in[/tex] g, h(z)[tex]\in[/tex] C since h[tex]\in[/tex] Isom C, which means h is an isometry from the complex plane to itself. QED.
Thanks in advance.