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.

     

Hypercentral units in integral group rings

In this note, we show that when is a torsion group the second center of the group of units of the integral group ring is generated by its torsion subgroup and by the center of . This extends a result of Arora and Passi (1993) from finite groups to torsion groups, and completes the characterization of hypercentral units in when is a torsion group.

     

Torsion units in integral group rings

Several special cases of the conjectures of Bovdi and Zassenhaus are proved. We also deal with special cases of the following conjecture: let α be a torsion unit of the integral group ring ZZG and m the smallest positive integer such that α^m ∈G then, m is a divisor of the exponent of the quotient group G/Z(G) provided this exponent is finite     

Nilpotent elements in group rings

The main theorem gives necessary and sufficient conditions for the rational group algebra QG to be without (nonzero) nilpotent elements if G is a nilpotent or F·C group. For finite groups G, a characterisation of group rings RG over a commutative ring with the same property is given. As an application those nilpotent or F·C groups are characterised which have the group of units U(KG) solvable for certain fields K.This work has been supported by N.R.C. Grant No. A-5300.     

Some intersections and identifications in integral group rings

LetZG be the integral group ring of a groupG and I(G) its augmentation ideal. For a free groupF andR a normal subgroup ofF, the intersectionI ⁿ⁺¹ (F) ∩I ⁿ⁺¹ (R) is determined for alln≥ 1. The subgroupsF ∩ (1+ZFI (R) I (F) I (S)) ANDF ∩ (1 + I (R)I ³ (F)) of F are identified whenR and S are arbitrary subgroups ofF.     

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.     

Generators of large subgroups of the unit group of integral group rings

This work is supported in part by NSERC Grant OGP0036631, Canada, and CNPq, Brasil     

Torsion units in integral group rings of solvable groups

A weaker version of the Zassenhaus conjecture for torsion units in integral group rings ZG is proved if G is either abelian-by-polycyclic or metabelian. As a consequence we obtain Bovdi's conjecture for torsion units in ZG for metabelian groups     

Torsion units in integral group rings of metacyclic groups

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 $\text{Z}$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 $\text{Z}$G of augmentation one and order relatively prime to |A| is rationally conjugate to an element of X.     

Lie properties of symmetric elements in group rings

Let ¹ be an involution of a group G extended linearly to the group algebra KG. We prove that if G contains no 2-elements and K is a field of characteristic $p\ne 2$, then the 1-symmetric elements of KG are Lie nilpotent (Lie n-Engel) if and only if KG is Lie nilpotent (Lie n-Engel).