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I'm trying to solve the following problem: Let $G$ be a metrizable compact topological group. Suppose for some $g_0 \in G$ the translation $L_{g_0}$ defined by $L_{g_0}g=g_0 g$ is topologically transitive. Then $G$ is an abelian group.

I know that since $L_{g_0}$ is topologically transitive, it is also minimal. I was hoping to show that the commutator of two elements is the identity and also wanted to incorporate the metrizability and compactness of the space somehow but didn't get too far.

Thanks

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Hints:

  1. Minimality means that every $L_{g_0}$-orbit is dense, in particular the orbit of the identity is dense.
  2. The orbit of the identity is just the powers of $g_0$.
  3. All powers of $g_0$ commute.
  4. Multiplication is continuous and you can approximate any two group elements with powers of $g_0$.
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+1 Just remembered I never got around to this one. –  user8126 Oct 2 '12 at 14:04
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