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     

Construction of free subgroups in the group of units of modular group algebras

Let KGbe the group algebra of a p¹ -group Gover a field Kof characteristic p > 0, and let U(KG)be its group of units. If KGcontains a nontrivial bicyclic unit and if Kis not algebraic over its prime field, then we prove that the free product Z_p? Z_p? Z_pcan be embedded in U(KG).     

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 .

     

On free subgroups of units of rings

We prove that if for elements of a ring of characteristic zero and is not nilpotent, then there exists such that the group generated by and is free nonabelian. This is used to prove that a noncommutative positive-definite algebra with involution over an uncountable field contains a free nonabelian subsemigroup.

     

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.

     

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     

On the torsion units of the integral group ring of finite projective special linear groups

H. J. Zassenhaus conjectured that any unit of finite-order and augmentation one in the integral group ring of a finite group G is conjugate in the rational group algebra to an element of G. One way to verify this is showing that such unit has the same distribution of partial augmentations as an element of G and the HeLP Method provides a tool to do that in some cases. In this paper, we use the HeLP Method to describe the partial augmentations of a hypothetical counterexample to the conjecture for the projective special linear groups.     

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     

On dense free subgroups of the automorphism group

We obtain results on existence of dense free subgroups of the automorphism group of a homogeneous model.     

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.