Central units of integral group rings of nilpotent groups
Eric Jespers1996, Proceedings of the American Mathematical Society
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Abstract
In this paper a finite set of generators is given for a subgroup of finite index in the group of central units of the integral group ring of a finitely generated nilpotent group. In this paper we construct explicitly a finite set of generators for a subgroup of finite index in the centre Z(U(ZG)) of the unit group U(ZG) of the integral group ring ZG of a finitely generated nilpotent group G. Ritter and Sehgal [4] did the same for finite groups G, giving generators which are a little more complicated. They also gave in [2] necessary and sufficient conditions for Z(U(ZG)) to be trivial; recall that the units ±G are called the trivial units. We first give a finite set of generators for a subgroup of finite index in Z(U(ZG)) when G is a finite nilpotent group. Next we consider an arbitrary finitely generated nilpotent group and prove that a central unit of ZG is a product of a trivial unit and a unit of ZT, where T is the torsion subgroup of G. As an application we obtain that the central units of ZG form a finitely generated group and we are able to give an explicit set of generators for a subgroup of finite index.
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