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1 results

  • Closed Symmetric Overgroups of Sn in On

  • Chi-Kwong Li, Wayne Whitney
  • Journal:
    Canadian Mathematical Bulletin / Volume 39 / Issue 1 / 01 March 1996
    Published online by Cambridge University Press:
    20 November 2018, pp. 83-94
    Print publication:
    01 March 1996
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  • A norm on ℝn is said to be permutation invariant if its value is preserved under permutation of the coordinates of a vector. The isometry group of such a norm must be closed, contain Sn and —I, and be conjugate to a subgroup of On, the orthogonal group. Motivated by this, we are interested in classifying all closed groups G such that 〈—I,Sn〉 < G < On. We use the theory of Lie groups to classify all possible infinite groups G, and use the theory of finite reflection groups to classify all possible finite groups G. In keeping with the original motivation, all groups arising are shown to be isometry groups. This completes the work of Gordon and Lewis, who studied the same problem and obtained the results for n ≥ 13.