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The structure of a group of permutation polynomials

Part of: Structure and classification of infinite or finite groups Permutation groups Metric geometry

Published online by Cambridge University Press:  09 April 2009

Gary L. Mullen  and
Harald Niederreiter
Gary L. Mullen
Affiliation:
Department of Mathematics The Pennsylvania State UniversityUniversity Park, Pennsylvania 16802, U.S.A.
Harald Niederreiter
Affiliation:
Mathematical InstituteAustrian Academy of Sciences Dr. Ignaz-Seipel-Platz 2 A-1010 Vienna, Austria
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Abstract

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Let Gq be the group of permutations of the finite field Fq of odd order q that can be represented by polynomials of the form ax(q+1)/2 + bx with a, b ∈ Fq. It is shown that Gq is isomorphic to the regular wreath product of two cyclic groups. The structure of Gq can also be described in terms of cyclic, dicyclic, and dihedral groups. It also turns out that Gq is isomorphic to the dymmetry group of a regular complex polygon.

MSC classification

Secondary: 20B25: Finite automorphism groups of algebraic, geometric, or combinatorial structures 20E22: Extensions, wreath products, and other compositions 51F15: Reflection groups, reflection geometries
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 38 , Issue 2 , April 1985 , pp. 164 - 170
Copyright
Copyright © Australian Mathematical Society 1985

References

[1]Carlitz, L., ‘Permutations in a finite field’, Proc. Amer. Math. Soc. 4 (1953), 538.CrossRefGoogle Scholar
[2]Cohen, A. M., ‘Finite complex reflection groups’, Ann. Sci. Ecole Norm. Sup. (4) 9 (1976), 379–436.CrossRefGoogle Scholar
[3]Coxeter, H. S. M., Regular complex polytopes (Cambridge Univ. Press, London, 1974).Google Scholar
[4]Coxeter, H. S. M. and Moser, W. O. J., Generators and relations for discrete groups (3rd ed., Springer-Verlag, Berlin-Heidelberg-New York, 1972).CrossRefGoogle Scholar
[5]Crowe, D. W., ‘The groups of regular complex polygons’, Canad. J. Math. 13 (1961), 149–156.CrossRefGoogle Scholar
[6]Fryer, K. D., ‘Note on permutations in a finite field’, Proc. Amer. Math. Soc. 6 (1955), 1–2.CrossRefGoogle Scholar
[7]Johnson, D. L., Presentation of groups (London Math. Soc. Lecture Note Series, Vol. 22, Cambridge Univ. Press, Cambridge, 1976).Google Scholar
[8]Lausch, H. and Nöbauer, W., Algebra of polynomials (North-Holland, Amsterdam, 1973).Google Scholar
[9]Lidl, R. and Niederreiter, H., Finite fields (Encyclopedia of Math. and Its Appl., Vol. 20, Addison-Wesley, Reading, Mass., 1983).Google Scholar
[10]Neumann, P. M., ‘On the structure of standard wreath products of groups’, Math. Z. 84 (1964), 343–373.CrossRefGoogle Scholar
[11]Niederreiter, H. and Robinson, K. H., ‘Complete mappings of finite fields’, J. Austral. Math. Soc. (Ser. A) 33 (1982), 197–212.CrossRefGoogle Scholar
[12]Nöbauer, W., ‘Über eine Klasse von Permutationspolynomen und die dadurch dargestellten Gruppen’, J. Reine Angew. Math. 231 (1968), 215–219.Google Scholar
[13]Shephard, G. C., ‘Regular complex polytopes’, Proc. London Math. Soc. (3) 2 (1952), 82–97.CrossRefGoogle Scholar
[14]Shephard, G. C., ‘Unitary groups generated by reflections’, Canad. J. Math. 5 (1953), 364–383.CrossRefGoogle Scholar
[15]Wells, C., ‘Groups of permutation polynomials’, Monatsh. Math. 71 (1967), 248–262.CrossRefGoogle Scholar
[16]Wells, C., ‘Generators for groups of permutation polynomials over finite fields’, Acta Sci. Math. Szeged 29 (1968), 167–176.Google Scholar