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On semigroups and groups of local polynomial functions

Published online by Cambridge University Press:  09 April 2009

Wilfried Nöbauer
Wilfried Nöbauer
Affiliation:
Institut für Algebra und Mathematische Strukturtheorie Technische UniversitätArgentinierstrasse 8 A-1040 Wien Austria
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Abstract

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Let Zn be the factor ring of the integers mod n and t be a positive integer. In this paper some results are given on the structure of the semigroup of all mappings from Zn into Zn and on the structure of the group of all permutations on Zn, which, for any t elements, can be represented by a polynomial function. If n = ab and a, b are relatively prime, then this (semi)group is isomorphic to the direct product of the respective (semi)groups for a and b. Thus it is sufficient to consider only the case where n = pe, p being a prime. In this case it is proved, that the (semi)group is isomorphic to the wreath product of a certain sub(semi)group of the symmetric (semi)group on Zpe−1 by the symmetric (semi)group on Zp. Some remarks on these sub(semi)groups are given.

Subject classification (Amer. Math. Soc. (MOS) 1970): 20 B 99, 13 B 25.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 28 , Issue 2 , September 1979 , pp. 214 - 218
Copyright
Copyright © Australian Mathematical Society 1979

References

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Lausch, H. and Nöbauer, W. (1979), ‘Local polynomial functions on factor rings of the integers’, J. Austral. Math. Soc., to appear.CrossRefGoogle Scholar
Nöbauer, W. (1974), ‘Compatible and conservative functions on residue-class rings of the integers’, Coll. Math. Soc. János Bolyai 13, Topics in number theory, 245–257.Google Scholar