UNITARY UNITS IN INTEGRAL GROUP RINGS

Abstract

0. Introduction In this article we continue the quest for subgroups of finite index in U(Z[G]) for G a finite group. Since the construction of finitely many generators for U(Z[G]) seems out of reach today, one might compromise by searching for a finite set of generators of a subgroup of finite index in the unit group. This question was stated for integral group rings as Problem 23 in [15]. When G is a finite abelian group, Bass and Milnor [2] constructed such a set of generators, namely the Bass cyclic units. Unfortunately, there are not many recipes for constructing units. Apart from the Bass cyclic units, there are for example the bicyclic units. Ritter and Sehgal [14] proved that the group generated by the Bass cyclic and bicyclic units (of the first type) is of finite index in U(Z[G]), when G is a finite nilpotent group of odd order. For a nilpotent group of even order this result is not valid in general. For example, in case of the non-abelian groups of order 16, which are not a Hamiltonian 2-group, this result only holds for D16 and D8 × C2 (using the bicyclics of both types) [6, 10, 11, 14].