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I hope someone could help me with the end of the proof. Maybe someone could give me a more detailed reason why this is true....

If the translation $L_{g_o}$ on a topological group is topologic transitiv, then it is minimal.

Let $g$ and $\hat{g}$ be in $G$. Set $A$ and $\hat{A}$ as the closures of the orbits. Now $g_0^n\hat{g} = g_0^ng(g^{-1}\hat{g})$. This is equiv. to $\hat{A} = Ag^{-1}\hat{g}$ and $\hat{A} = G \Leftrightarrow A = G$.

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