Constructing free subgroups of integral group ring units

Let be an arbitrary group. It is proved that if contains a bicyclic unit , then is a nonabelian free subgroup of invertible elements.

     

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     

Central units of the integral group ring

There are very few cases known of nonabelian groups where the group of central units of , denoted , is nontrivial and where the structure of , including a complete set of generators, has been determined. In this note, we show that the central units of augmentation 1 in the integral group ring form an infinite cyclic group , and we explicitly find the generator .

     

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.     

Finitely generated group ring units

We give a classification of nilpotent groups for which the unit group of the integral group ring is finitely generated.

     

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     

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.

     

On the geaded ring associated with an integral group ring

Let G be the direct product of cyclic groups of order 4. we calculate an upper bound for the degree of the summands of gr ?G. There are grounds for believing that this bound is, in fact, the exact value in all cases.