Let $G$ be a complex group, i.e. a complex manifold with a group structure. Assume that for each $g\in G$, the map $$G\rightarrow G,h\mapsto gh^{-1}$$ is holomorphic. Can it imply that $G$ is a complex Lie group? Or equivlently, is the map $$G\times G\rightarrow G,(g,h)\mapsto gh^{-1}$$ also holomorphic?
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