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