On the torsion units in integral group rings of dihedral 2-groups

A. Weiss proved in [16] that for a p-group G and a nite subgroup U of the units of augmentation 1 of the group ring ẐZpG over the p-adic integers there exists a group basis containing U. In [12] K. W. Roggenkamp and L. L. Scott proved that all group bases in ẐZpG are conjugate. We will be concerned with the analogous question in ZZG, the integral group ring of a p-group G: Are nite subgroups of the units of augmentation 1 of ZZG subgroups of group bases? I. Hughes and K. E. Pearson showed in [8] that in the group of units of augmentation 1 of the integral group ring of the dihedral group of order 6 there is an involution not being part of any group basis. This involution however is not even part of a group basis in the group ring over the 2-adic integers. Therefore, they measure a local phenomenon rather than a global one. Furthermore, the dihedral group of order 6 is not a p-group and A. Weiss' theorem is not available. We investigate this problem for dihedral groups of order a power of 2 such that A. Weiss ' theorem can be applied. To go into details: Let D2n be the dihedral group of order 2 n+1. We denote by V (RG) the group of units of augmentation 1 in the group ring of the nite group G over the integral domain R. The augmentation is the map induced by the trivial representation. As announced in [18] we will show that in ZZD2n the number of conjugacy classes of involutions in V (ZZD2n) is equal to 1 + 2 2n1 nY k=1 h