[math] \newcommand{\mathds}{\mathbb}[/math]

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Prove that the compact quantum groups [math]G[/math] which are finite, in the sense that [math]\dim C(G) \lt \infty[/math], coincide with the discrete quantum groups [math]\Gamma[/math] which are finite, in the sense that [math]\dim C^*(\Gamma) \lt \infty[/math], and coincide as well with the finite quantum groups.