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Affine Completeness of Generalised Dihedral Groups

  • Jürgen Ecker (a1)
Abstract

In this paper we study affine completeness of generalised dihedral groups. We give a formula for the number of unary compatible functions on these groups, and we characterise for every k ∈ N the k-affine complete generalised dihedral groups. We find that the direct product of a 1-affine complete group with itself need not be 1-affine complete. Finally, we give an example of a nonabelian solvable affine complete group. For nilpotent groups we find a strong necessary condition for 2-affine completeness.

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References
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[1] Aichinger, E., 2-affine complete algebras need not be affine complete. Algebra Universalis 47(2002), no. 4, 425434.
[2] Aichinger, E., The polynomial functions on certain semidirect products of groups. Acta Sci. Math. (Szeged) 68(2002), no. 1–2, 6381.
[3] Ecker, J., On the number of polynomial functions on nilpotent groups of class 2. In: Contributions to General Algebra 10, Heyn, Klagenfurt, 1998, pp. 133137.
[4] Hall, P., The Edmonton Notes on Nilpotent Groups. Queen Mary College Mathematics Notes, Mathematics Department, Queen Mary College, London, 1969.
[5] Lausch, H. and Nöbauer, W., Algebra of Polynomials. North-Holland Mathematical Library 5, North-Holland, Amsterdam, 1973.
[6] Lausch, H. and Nöbauer, W., Funktionen auf endlichen Gruppen. Publ. Math. Debrecen 23(1976), no. 1–2, 5361.
[7] Lyons, C. G. and Mason, G., Endomorphism near-rings of dicyclic and generalised dihedral groups. Proc. Roy. Irish Acad. Sect. A 91(1991), no. 1, 99111.
[8] Nöbauer, W., Über die affin vollständigen, endlich erzeugbaren Moduln. Monatsh. Math. 82(1976), no. 3, 187198.
[9] Robinson, D. J. S., A Course in the Theory of Groups. Second edition. Graduate Texts in Mathematics 80, Springer-Verlag, New York, 1996.
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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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