Eric Jespers

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.

On Symmetric Elements and Symmetric Units in Group Rings

ABSTRACT Let R be a commutative ring, G a group, and RG its group ring. Let ϕ: RG → RG denote the R-linear extension of an involution ϕ defined on G. An element x in RG is said to be symmetric if ϕ (x) = x. A characterization is given of when the symmetric elements (RG)ϕ of RG form a ring. For many domains R it is also shown that (RG)ϕ is a ring if and only if the symmetric units form a group. The results obtained extend earlier work of Bovdi (2001), Bovdi et al. (1996), Bovdi and Parmenter (1997), Broche Cristo (2003, to appear), Giambruno and Sehgal (1993), and Lee (1999), who dealt with the case that ϕ is the involution * mapping g ∈ G onto g−1.

Torsion units in integral group rings of Janko simple groups

Using the Luthar-Passi method, we investigate the classical Zassenhaus conjecture for the normalized unit group of integral group rings of Janko sporadic simple groups. As a consequence, we obtain that the Gruenberg-Kegel graph of the Janko groups J 1 , J 2 and J 3 is the same as that of the normalized unit group of their respective integral group ring.

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. In this paper we construct explicitly a finite set of generators for a subgroup of finite index in the centre Z(U (ZG)) of the unit group U (ZG) of the integral group ring ZG of a finitely generated nilpotent group G. Ritter and Sehgal [4] did the same for finite groups G, giving generators which are a little more complicated. They also gave in [2] necessary and sufficient conditions for Z(U(ZG)) to be trivial; recall that the units ::i:G are called the trivial units. We first give a finite set of generators for a subgroup of finite index in Z (U(ZG)) when G is a finite nilpotent group. Next we consider an arbitrary finitely generated nilpotent group and prove that a central unit of ZG is a product of a trivial unit and a unit of ZT, where T is the torsion subgroup of G. As an application we obtain that the central units of ZG form a finitely generated group and we are able to give an explicit set of generators for a subgroup of finite index. 1. FINITE NILPOTENTGROUPS Throughout this section G is a finite group. When G is Abelian, it was shown in [1] that the Bass cyclic units generate a subgroup of finite index in the unit group. Using a stronger version of this result, also proved by Bass in [1], we will construct a finite set of generators from the Bass cyclic units when G is finite nilpotent. Our notation will follow that in [6]. The following lemma is proved in [1]. Lemma 1. The images of the Bass cyclic units of ZG under the natural homomorphism j : U(ZG) -+ K1 (ZG) generate a subgroup of finite index. Let L denote the kernel of this map j, and B the subgroup of U(ZG) generated by the Bass cyclic units. It follows that there exists an integer m such that zm E LB for all z E Z(U(ZG)), and so we can write zm = Ib1b2 . .. bk for some 1 ELand Bass cyclic units bi. Received by the editors August 4, 1994. 1991 Mathematics Subject Classification. Primary 16U60, 20CO5, 20CO7; Secondary 20C10, 20C12. This work is supported in part by NSERC Grants OGPOO36631, A8775 and A5300, Canada, and by DGICYT, Spain. @1996 American Mathematical Society

CENTRAL IDEMPOTENTS IN THE RATIONAL GROUP ALGEBRA OF A FINITE NILPOTENT GROUP

We describe the primitive central idempotents of a rational group algebra of a finite nilpotent group. The description does not make use of the character table of the group G.

ANTISYMMETRIC ELEMENTS IN GROUP RINGS II

Research supported by Onderzoeksraad of Vrije Universiteit Brussel, Fonds voor Wetenschappelijk Onderzoek (Vlaanderen), Flemish-Polish bilateral agreement BIL 01/31, FAPEMIG and CNPq. Proc. 300243/79-0(RN) of Brazil, D.G.I. of Spain and Fundacion Seneca of Region de Murcia.

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

We give an explicit and character-free construction of a complete set of orthogonal primitive idempotents of a rational group algebra of a finite nilpotent group and a full description of the Wedderburn decomposition of such algebras. An immediate consequence is a well-known result of Roquette on the Schur indices of the simple components of group algebras of finite nilpotent groups. As an application, we obtain that the unit group of the integral group ring

Submonoids of polycyclic-by-finite groups and their algebras

{\mathbb Z} G

Units of group rings of groups of order 16

of a finite nilpotent group G has a subgroup of finite index that is generated by three nilpotent groups for which we have an explicit description of their generators. Another application is a new construction of free subgroups in the unit group. In all the constructions dealt with, pairs of subgroups (H, K), called strong Shoda pairs, and explicit constructed central elements e(G, H, K) play a crucial role. For arbitrary finite groups we prove that the primitive central idempotents of the rational group algebras are rational linear combinations of such e(G, H, K), with (H, K) strong Shoda pairs in subgroups of G.

Free Groups and Subgroups of Finite Index in the Unit Group of an Integral Group Ring

We describe Noetherian semigroup algebras K[S] of submonoids S of polycyclic-by-finite groups over a field K. As an application, we show that these algebras are finitely presented and also that they are Jacobson rings. Next we show that every prime ideal P of K[S] is strongly related to a prime ideal of the group algebra of a subgroup of the quotient group of S via a generalised matrix ring structure on K[S]/P. Applications to the classical Krull dimension, prime spectrum, and irreducible K[S]-modules are given.