Hostname: page-component-77c89778f8-vsgnj Total loading time: 0 Render date: 2024-07-23T00:09:38.166Z Has data issue: false hasContentIssue false
  • English
  • Français

Affine Completeness of Generalised Dihedral Groups

Published online by Cambridge University Press:  20 November 2018

Jürgen Ecker
Jürgen Ecker*
Affiliation:
Institut für Algebra, Johannes Kepler Universität Linz, 4040 Linz, Austria and Computer- und Mediensicherheit, FH Oö Campus Hagenberg, 4232 Hagenberg, Austria e-mail: juergen.ecker@fh-hagenberg.at
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 $k\in \mathbb{N}$ 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.

Keywords

08A4016Y3020F05
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 49 , Issue 3 , 01 September 2006 , pp. 347 - 357
Copyright
Copyright © Canadian Mathematical Society 2006

References

[1] Aichinger, E., 2-affine complete algebras need not be affine complete. Algebra Universalis 47(2002), no. 4, 425–434.Google Scholar
[2] Aichinger, E., The polynomial functions on certain semidirect products of groups. Acta Sci. Math. (Szeged) 68(2002), no. 1–2, 63–81.Google Scholar
[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. 133–137.Google Scholar
[4] Hall, P., The Edmonton Notes on Nilpotent Groups. Queen Mary College Mathematics Notes, Mathematics Department, Queen Mary College, London, 1969.Google Scholar
[5] Lausch, H. and Nöbauer, W., Algebra of Polynomials. North-Holland Mathematical Library 5, North-Holland, Amsterdam, 1973.Google Scholar
[6] Lausch, H. and Nöbauer, W., Funktionen auf endlichen Gruppen. Publ. Math. Debrecen 23(1976), no. 1–2, 53–61.Google Scholar
[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, 99–111.Google Scholar
[8] Nöbauer, W., Über die affin vollständigen, endlich erzeugbaren Moduln. Monatsh. Math. 82(1976), no. 3, 187–198.Google Scholar
[9] Robinson, D. J. S., A Course in the Theory of Groups. Second edition. Graduate Texts in Mathematics 80, Springer-Verlag, New York, 1996.Google Scholar