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Józiak, Paweł
 Remarks on Hopf images and quantum permutation groups $S_n^+$ Motivated by a question of A. Skalski and P.M. SoÅtan (2016) about inner faithfulness of the S. Curran's map of extending a quantum increasing sequence to a quantum permutation, we revisit the results and techniques of T. Banica and J. Bichon (2009) and study some group-theoretic properties of the quantum permutation group on $4$ points. This enables us not only to answer the aforementioned question in positive in case $n=4, k=2$, but also to classify the automorphisms of $S_4^+$, describe all the embeddings $O_{-1}(2)\subset S_4^+$ and show that all the copies of $O_{-1}(2)$ inside $S_4^+$ are conjugate. We then use these results to show that the converse to the criterion we applied to answer the aforementioned question is not valid. Keywords:Hopf image, quantum permutation group, compact quantum groupCategories:20G42, 81R50, 46L89, 16W35
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