César Polcino Milies

Idempotents in group algebras and minimal abelian codes

We compute the number of simple components of a semisimple finite abelian group algebra and determine all cases where this number is minimal; i.e. equal to the number of simple components of the rational group algebra of the same group. This result is used to compute idempotent generators of minimal abelian codes, extending results of Arora and Pruthi [S.K. Arora, M. Pruthi, Minimal cyclic codes of length 2p^n, Finite Field Appl. 5 (1999) 177-187; M. Pruthi, S.K. Arora, Minimal codes of prime power length, Finite Field Appl. 3 (1997) 99-113]. We also show how to compute the dimension and weight of these codes in a simple way.

The normalizer property for integral group rings of Frobenius groups

We prove that the normalizer property holds for the integral group ring of a finite Frobenius group G; i.e., that the normalizer of G in the group of units of its integral group ring is NU (G) = G · ζ ,w hereζ denotes the centre of the unit group.

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.

Symmetric Elements Under Oriented Involutions in Group Rings

Let RG denote the group ring of a group G over a commutative ring with unity R. Given a homomorphism σ:G → {± 1} and an involution ϕ of the group G, an oriented invoultion σϕ of RG is defined in a natural way. We characterize when the set of symmetric elements under this involution is a subring. This gives a unified setting for earlier work of several authors.

Torsion units in alternative loop rings

Let ZL denote the integral alternative loop ring of a finite loop L. If L is an abelian group, a well-known result of G. Higman says that ±g,g € L are the only torsion units (invertible elements of finite order) in ZL . When L is not abelian, another obvious source of units is the set ±y~l gy of conjugates of elements of L by invertible elements in the rational loop algebra QL . H. Zassenhaus has conjectured that all the torsion units in an integral group ring are of this form. In the alternative but not associative case, one can form potentially more torsion units by considering conjugates of conjugates V^\y7(g7l)V\ and so forth. In this paper we prove that every torsion unit in an alternative loop ring over Z is ± a conjugate of a conjugate of a loop element.

Integral group rings with nilpotent unit groups

Introduction. LetR be a ring with unit element and G a finite group. We denote by RG the group ring of the group G over R and by U(RG) the group of units of this group ring. The study of the nilpotency of U{RG) has been the subject of several papers. First, J. M. Bateman and D. B. Coleman showed in [1] that if G is a finite group and K a field, then U(KG) is nilpotent if and only if either char K = 0 and G is abelian or char K = p ^ 0 and G is the direct product of a ^-group and an abelian group. Later K. Motose and H. Tominaga [6] corrected a small gap in the proof of the theorem above and obtained a similar result for group rings of finite groups over artinian semisimple rings (which must be commutative for U(RG) to be nilpotent). For group rings over commutative rings of non-zero characteristic it is possible to obtain a natural generalization of the theorem in [1]. (See I . I . Khripta [5] or C. Polcino [7]). In this paper we study the nilpotency of U(ZG) where Z is the ring of rational integers. In Section 2 we consider also group rings over rings of £-adic integers. A brief account of the results in that section was given in [7].

Group rings whose symmetric units are nilpotent

Abstract Let K be an infinite field of characteristic different from 2, and G a group containing elements of infinite order. We classify the groups G such that the symmetric units of KG satisfy the identity (x 1, x 2, …, x n ) = 1, for some n.

Integral group rings of frobenius groups and the conjectures of H.J. Zassenhaus

The conjecture of H.J. Zassenhaus for finite subgroups of units of integral group rings. restricted to p-subgroups, is proved for finite Frobenius groups when p is an odd prime. The result for 2-subgroups is established for those Frobenius groups that cannot be mapped homo-morphically onto S

Isomorphisms of integral alternative loop rings

sub:5

Lie Properties of Symmetric Elements Under Oriented Involutions

esub:. The conjecture in its full strength is proved for A5, S5 and SL(2.5).