Ángel del Río

On Monomial Characters and Central Idempotents of Rational Group Algebras

Abstract We give a method to obtain the primitive central idempotent of the rational group algebra ℚG over a finite group G associated to a monomial irreducible character which does not involve computations with the character field nor its Galois group. We also show that for abelian-by-supersolvable groups this method takes a particularly easy form that can be used to compute the Wedderburn decomposition of ℚG.

Involutive Yang-Baxter groups

In 1992 Drinfeld posed the question of finding the set-theoretic solutions of the Yang-Baxter equation. Recently, Gateva-Ivanova and Van den Bergh and Etingof, Schedler and Soloviev have shown a group-theoretical interpretation of involutive non-degenerate solutions. Namely, there is a one-to-one correspondence between involutive non-degenerate solutions on finite sets and groups of I-type. A group G of I-type is a group isomorphic to a subgroup of Fa n ⋊ Sym n so that the projection onto the first component is a bijective map, where Fa n is the free abelian group of rank n and Sym n is the symmetric group of degree n. The projection of G onto the second component Sym n we call an involutive Yang-Baxter group (IYB group). This suggests the following strategy to attack Drinfelds problem for involutive non-degenerate set-theoretic solutions. First classify the IYB groups and second, for a given IYB group G, classify the groups of I-type with G as associated IYB group. It is known that every IYB group is solvable. In this paper some results supporting the converse of this property are obtained. More precisely, we show that some classes of groups are IYB groups. We also give a non-obvious method to construct infinitely many groups of I-type (and hence infinitely many involutive non-degenerate set-theoretic solutions of the Yang-Baxter equation) with a prescribed associated IYB group.

Zassenhaus Conjecture for cyclic-by-abelian groups

Zassenhaus Conjecture for torsion units states that every aug- mentation one torsion unit of the integral group ring of a finite group G is conjugate to an element of G in the units of rational group algebra QG. This conjecture has been proved for nilpotent groups, metacyclic groups and some other families of groups. We prove the conjecture for cyclic-by-abelian groups. In this paper G is a finite group and RG denotes the group ring of G with coefficients in a ringR. The units of RG of augmentation one are usually called normalized units. In the 1960s Hans Zassenhaus established a series of conjectures about the finite subgroups of normalized units of ZG. Namely he conjectured that every finite group of normalized units of ZG is conjugate to a subgroup of G in the units of QG. These conjecture is usually denoted (ZC3), while the version of (ZC3) for the particular case of subgroups of normalized units with the same cardinality as G is usually denoted (ZC2). These conjectures have important consequences. For example, a positive solution of (ZC2) implies a positive solution for the Isomorphism and Automorphism Problems (see (Seh93) for details). The most celebrated positive result for Zassenhaus Conjectures is due to Weiss (Wei91) who proved (ZC3) for nilpotent groups. However Roggenkamp and Scott founded a counterexample to the Automorphism Problem, and henceforth to (ZC2) (see (Rog91) and (Kli91)). Later Hertweck (Her01) provided a counterexample to the Isomorphism Problem. The only conjecture of Zassenhaus that is still up is the version for cyclic sub- groups namely:

Globalizations of partial actions on nonunital rings

In this note we prove a criteria for the existence of a globalization for a given partial action of a group on an s-unital ring. If the globalization exists, it is unique in a natural sense. This extends the globalization theorem from Dokuchaev and Exel, 2005, obtained in the context of rings with 1.

Graded rings and equivalences of categories

(1991). Graded rings and equivalences of categories. Communications in Algebra: Vol. 19, No. 3, pp. 997-1012.

An intrinsical description of group codes

A (left) group code of length n is a linear code which is the image of a (left) ideal of a group algebra via an isomorphism

Rational Group Algebras of Finite Groups: From Idempotents to Units of Integral Group Rings

{\mathbb{F}G \rightarrow \mathbb{F}^n}

The picard group of a category of graded modules

which maps G to the standard basis of