The Symmetric, Alternating, and other Special Groups

  • H. S. M. Coxeter
  • W. O. J. Moser
Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete book series (MATHE1, volume 14)


The importance of the symmetric group is obvious from the fact that any finite group, of order n, say, is a subgroup of S n . The procedure for obtaining generators and relations for S n can be carried over almost unchanged for a certain infinite group, first studied by Artin (1926); accordingly, we begin by describing this so-called braid group. The symmetric group (§ 6.2, p. 64) has a subgroup An, of index 2 (§ 6.3, p. 66), which is particularly interesting because, when n > 4, it is simple. In § 6.4, p. 67, we exhibit the groups S3, A4, S4 and A5 as members of the family of polyhedral groups (l, m, n),defined by


Fundamental Group Symmetric Group Factor Group Braid Group Fundamental Region 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1957

Authors and Affiliations

  • H. S. M. Coxeter
    • 1
  • W. O. J. Moser
    • 2
  1. 1.University of TorontoCanada
  2. 2.University of SaskatchewanCanada

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