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

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Futher advance in your understanding of [math]S_N\to S_N^+[/math], as follows:

• Prove that [math]S_3^+=S_3[/math], by using a clever method, of your choice.
• Prove that [math]S_4^+\neq S_4[/math], again by using a clever method, of your choice.
• Prove that [math]S_4^+[/math] is coamenable, while [math]S_5^+[/math] is not coamenable.
• Can we talk about quantum permutations of finite quantum spaces?
• If yes, can you prove that for [math]M_2[/math], given by [math]C(M_2)=M_2(\mathbb C)[/math], we get [math]SO_3[/math]?
• Based on this, can we say that [math]S_4^+[/math] should be a kind of twist of [math]SO_3[/math]?