Central Units of Integral Group Rings of Nilpotent Groups

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. 1. Finite nilpotent groups Throughout this section G is a finite group. When G is Abelian, it was shown in [1] that the Bass cyclic units generate a subgroup of finite index in the unit group. Using a stronger version of this result, also proved by Bass in [1], we will construct a finite set of generators from the Bass cyclic units when G is finite nilpotent. Our notation will follow that in [6]. The following lemma is proved in [1]. Lemma 1. The images of the Bass cyclic units of ZG under the natural homomorphism j : U(ZG)→ K1(ZG) generate a subgroup of finite index. Let L denote the kernel of this map j, and B the subgroup of U(ZG) generated by the Bass cyclic units. It follows that there exists an integer m such that z ∈ LB for all z ∈ Z(U(ZG)), and so we can write z = lb1b2 · · · bk for some l ∈ L and Bass cyclic units bi. Received by the editors August 4, 1994. 1991 Mathematics Subject Classification. Primary 16U60, 20C05, 20C07; Secondary 20C10, 20C12. This work is supported in part by NSERC Grants OGP0036631, A8775 and A5300, Canada, and by DGICYT, Spain. c ©1996 American Mathematical Society 1007 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 1008 E. JESPERS, M. M. PARMENTER, AND S. K. SEHGAL Next, let Zi denote the i-th centre of G, and suppose from now on that G is nilpotent of class n. For any x ∈ G and Bass cyclic unit b ∈ Z〈x〉, we define b(1) = b and for 2 ≤ i ≤ n b(i) = ∏ g∈Zi bg(i−1), where α = g−1αg for α ∈ ZG. Note that by induction b(i) is central in Z〈Zi, x〉 and independent of the order of the conjugates in the product expression. In particular, b(n) ∈ Z(U(ZG)). Recall again that if z ∈ Z(U(ZG)), then z = lb1b2 · · · bk for some l ∈ L and Bass cyclic units bi. Since K1(ZG) is Abelian, we can write zm|Z2||Z3|···|Zn| = (lb1b2 · · · bk)23n = l1 ∏ 1≤i≤k b |Z2||Z3|···|Zn| i for some l1 ∈ L = l2 ∏ 1≤i≤k b |Z3|···|Zn| i(2) for some l2 ∈ L = l′ ∏ 1≤i≤k bi(n) for some l ′ ∈ L. Since each bi(n) is in Z(U(ZG)), we conclude that l′ ∈ L ∩ Z(U(ZG)). But we shall show next that L ∩ Z(U(ZG)) is trivial, so l′ ∈ ±Z(G). The argument uses the same idea as in [3, Lemma 3.2]. For every primitive central idempotent e in the rational group algebra QG, the simple ring QGe has a reduced norm which we denote by nre. Further, denote me = √ [QGe : Z(QGe)]