I would like to show that they are subsets of each other. I have shown that $\mathcal{O}_{(0,0)} \subseteq \{(0, 0)\}$ but I need help showing the other way please.
2026-03-29 15:20:15.1774797615
Prove that $\mathcal{O}_{(0,0)} = \{(0, 0)\}.$ The action is given by $\lambda \cdot (x, y) = (\lambda x, \lambda y).$
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Since $\lambda\cdot (x, y)$ is an action, the orbit of each element $(a, b)$ contains $(a,b)$ by letting $\lambda=e$. Can you conclude the result from this?