Polytopes of high rank for the symmetric groups |
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Authors: | Maria Elisa Fernandes Dimitri Leemans |
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Affiliation: | aCenter for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, Aveiro, Portugal;bUniversité Libre de Bruxelles, C.P.216, Géométrie, Boulevard du Triomphe, B-1050 Bruxelles, Belgium |
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Abstract: | In the Atlas of abstract regular polytopes for small almost simple groups by Leemans and Vauthier, the polytopes whose automorphism group is a symmetric group Sn of degree 5?n?9 are available. Two observations arise when we look at the results: (1) for n?5, the (n−1)-simplex is, up to isomorphism, the unique regular (n−1)-polytope having Sn as automorphism group and, (2) for n?7, there exists, up to isomorphism and duality, a unique regular (n−2)-polytope whose automorphism group is Sn. We prove that (1) is true for n≠4 and (2) is true for n?7. Finally, we also prove that Sn acts regularly on at least one abstract polytope of rank r for every 3?r?n−1. |
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Keywords: | Polytopes Permutation groups Independent generating sets C-groups |
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