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A Geometric Model for the Generalized Symmetric Group

Published online by Cambridge University Press:  20 November 2018

Norman W. Johnson
Norman W. Johnson*
Affiliation:
University of Toronto
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The generalized symmetric group S(n, m) consists of all permutations of mn symbols commutative with

Since each cycle Qi = (1i 2i … mi) is of order m, there are mn permutations within the n cycles, generating an invariant subgroup Q of order mn. Also, there are n! ways of permuting the cycles among themselves, by transformations

where i1, i2, …, in are the symbols 1, 2, …, n in some order [5, p. 39]. The permutations W* form a subgroup Sn* of order n!, isomorphic to the symmetric group Sn.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 3 , Issue 2 , 01 May 1960 , pp. 133 - 142
Copyright
Copyright © Canadian Mathematical Society 1960

References

1. Coxeter, H. S. M., The polytopes with regular-prismatic vertex figures, Philos. Trans. Roy. Soc. London Ser. A 229 (1930), 329-425.Google Scholar
2. Coxeter, H. S. M., The abstract groups Rm=Sm=(RjSj)Pj = l, sm=T2=(SJT)2Pj = l, and Sm=T2=(S-JTSJT)PJ=1, Proc. London Math. Soc. (2) 41 (1936), 278-301.Google Scholar
3. Coxeter, H. S. M., Regular Polytopes (London, 1948; New York, 1949).Google Scholar
4. Manning, H. P., Geometry of Four Dimensions (New York, 1914).Google Scholar
5. Osima, M., On the representations of the generalized symmetric group, Math. J. Okayama Univ. 4 (1954), 39-56.Google Scholar
6. Shephard, G. C., Regular complex polytopes, Proc. London Math. Soc. (3) 2 (1952), 82-97.Google Scholar
7. Shephard, G. C., Unitary groups generated by reflections, Canad. J. Math. 5 (1953), 364-383.Google Scholar
8. Shephard, G. C. and Todd, J. A., Finite unitary reflection groups, Canad. J. Math. 6 (1954), 274-304.10.4153/CJM-1954-028-3Google Scholar