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     

Certain conjugacy classes of units in integral group rings of metacyclic groups

Let G be the metacyclic group of order pq given by

$\text{G = 〈σ, τ: σ}{}^{\mathrm{p}}\text{= 1 = τ}{}^{\mathrm{q}}\text{, τστ}{}^{\mathrm{?}}\text{= σ}{}^{\mathrm{j}}\text{〉}$

where p is an odd prime, q ≥ 2 a divisor of p ? 1, and where j belongs to the exponent q mod p. Let V denote the group of units of augmentation 1 in the integral group ring $\text{Z}$G of G. In this paper it is proved that the number of conjugacy classes of elements of order p in V is

$\text{(p ? 1)}{}^{\mathrm{q?1}}\text{μ}{}_0\text{H}\text{vq}$

where ν, μ₀, and H are suitably defined numbers.     

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     

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

We prove that any torsion unit of the integral group ring ZG 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.     

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.

     

Groups of units of integral group rings commensurable with direct products of free-by-free groups

We classify the finite groups G such that the group of units of the integral group ring ZG has a subgroup of finite index which is a direct product of free-by-free groups.     

Torsion units in the integral group ring of S4

Any torsion unit inZZS ₄ is rationally conjugate to a trivial unit. This work was supported by CAPES of Brazil     

Describing units of integral group rings of some 2-groups

Let R be a ring and M a right R-module. The module M is a CS-module or satifies (C₁) if every submodule is essential in a direct summand of M. In this note we investigate two generalizations of CS-modules.     

Groups of units of integral group rings of Kleinian type

We explore a method to obtain presentations of the group of units of an integral group ring of some finite groups by using methods on Kleinian groups. We classify the nilpotent finite groups with central commutator for which the method works and apply the method for two concrete groups of order 16.

     

Free products of abelian groups in the unit group of integral group rings

We classify finite groups which are such that the unit group of the integral group ring has a subgroup of finite index which is a non-trivial free product of abelian groups.

     

Zassenhaus Conjecture (ZC1) on torsion units of integral group rings for some metabelian groups

We prove a conjecture of Zassenhaus that every normalized torsion unit of the integral group ring ZG of a finite group G is rationally conjugate to a group element for some metabelian groups including metacyclic groups G containing a normal cyclic group A such that G/A is cyclic of prime power order. The relative prime case was done in [11]. Received: 21 April 2005