Free Pairs of Bicyclic Units in the Integral Group Ring of the Dihedral Group

Abstract

We prove that if D _2n is the dihedral group of order 2n and u and v are two noncommuting bicyclic units in ℤD _2n of the same type, then the pair {u, v} always generates a free group of rank 2 except possibly if 12 divides n. The problematic cases are explicitly described. Further, it is shown that a problematic pair {u, v} generates a free group if and only if is a free point and that {u, v ²} always generates a free group. Thus we answer in the affirmative a conjecture by Jespers et al. (2002).

Key Words:

• Bicyclic unit
• Dihedral group
• Integral group ring

2000 Mathematics Subject Classification:

• Primary 16S34
• 16U60

ACKNOWLEDGMENTS

This work has been partially supported by MEC (Ministerio de Educación y Ciencia, Spain) and FEDER (Fondo Europeo de Desarrollo Regional), grant MTM2005-03868, and Fundación Séneca (Comunidad Autónoma de la Región de Murcia, Spain), grant 00684/PI/04.

We are indebted to Professor Á. del Río for his enlightening remarks on the first part of the article.

Notes

Communicated by M. Ferrero.