Sudarshan K. Sehgal

Torsion units in integral group rings of some metabelian groups

Let G be an extension of an elementary abelian p-group A by an abelian group X with faithful and irreducible action. Then any torsion unit of the integral group ring ZG is rationally conjugate to a trivial unit. This confirms a conjecture of Zassenhaus for these groups G.

Torsion units in integral group rings of some metabelian groups, II

Abstract We prove that any torsion unit of the integral group ring Z G is rationally conjugate to a trivial unit if G = A ⋊ X with both A and X abelian, | Xz . sfnc ; p for every prime p dividing | A | provided either | X | is prime or A ic cyclic.

Lie nilpotence of group rings

Let FG be the group algebra of a group G over a field F. Denote by ∗ the natural involution, (∑fi gi -1. Let S and K denote the set of symmetric and skew symmetric and skew symmetric elements respectively with respect to this involutin. It is proved that if the characteristic of F is zero p≠2 and G has no 2-elements, then the Lie nilpotence of S or K implies the Lie nilpotence of FG.

Group algebras whose units satisfy a group identity

Let FG be the group algebra of a torsion group over an infinite field F . Let U be the group of units of FG. We prove that if U satisfies a group identity, then FG satisfies a polynomial identity. This confirms a conjecture of Brian Hartley.

Units of integral group rings of metabelian groups

Abstract Let U1( Z G) denote the units of augmentation one of the integral group ring Z G of the finite group G. We prove that G has a normal torsion-free complement in U1( Z G) if G has an abelian normal subgroup A and G A is abelian, of odd order, or of exponent dividing 4 or 6.

On the normalizer property for integral group rings

Let R(G) denote the intersection of all nonnormal subgroups of a group G. In this note, we prove that for every finite group G, if R(G) is not trivial, then the normalizer property holds forG.

Constructing free subgroups of integral group ring units

Let G be an arbitrary group. It is proved that if ZG contains a bicyclic unit u 6= 1, then 〈u, u∗〉 is a nonabelian free subgroup of invertible elements.

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 ::i: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 NILPOTENTGROUPS 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 zm E LB for all z E Z(U(ZG)), and so we can write zm = Ib1b2 . .. bk for some 1 ELand Bass cyclic units bi. Received by the editors August 4, 1994. 1991 Mathematics Subject Classification. Primary 16U60, 20CO5, 20CO7; Secondary 20C10, 20C12. This work is supported in part by NSERC Grants OGPOO36631, A8775 and A5300, Canada, and by DGICYT, Spain. @1996 American Mathematical Society

Integral group rings with trivial central units

Let It = %{ZG) be the unit group of the integral group ring ZC. It is a classical result of G. Higman (Sehgal, 1978, p. 57) that if G is torsion then IL is trivial if and only if G is abelian of exponent 2, 3, 4 or 6 or G = ^s x E, the product of the quaternion group K^ of order 8 and an elementary abelian 2-group E. Since triviality of IL for torsion free groups G is an open problem it is quite difficult to extend this result to arbitrary groups. However, if a mild condition is imposed on G a classification can be seen from the results in Sehgal (1978) as we show in Sec. 5. In 1990, Ritter and Sehgal classified finite groups G so that the central units of ZG are trivial, namely of the form ±g, where g is an element of the centre of G. A classification of arbitrary groups G with trivial central units in ZG was asked for in Sehgal (1993, Problem 26, p. 301). Recently, Parmenter (1999) renewed this call. We give below this classification in terms of the condition of Ritter and Sehgal (1990).

Torsion Units in Integral Group Rings of Metacyclic Groups

Abstract It is proved that if G is a split extension of a cyclic p-group by a cyclic p′-group with faithful action then any torsion unit of augmentation one of Z G is rationally conjugate to a group element. It is also proved that if G is a split extension of an abelian group A by an abelian group X with (|A|, |X|) = 1 then any torsion unit of Z G of augmentation one and order relatively prime to |A| is rationally conjugate to an element of X.