Group algebras whose units satisfy a group identity (original) (raw)
Group Identities on Units of Group Algebras
Journal of Algebra, 2000
Let U be the group of units of the group algebra FG of a group G over a field F. Suppose that either F is infinite or G has an element of infinite order. We characterize groups G so that U satisfies a group identity. Under the assumption that G modulo the torsion elements is nilpotent this gives a complete classification of such groups. For torsion groups this problem has already been settled in recent years. @ 2000AcademicPress .
Star-group identities on units of group algebras: The non-torsion case
Forum Mathematicum, 2017
Let G be a group, F a field and FG the corresponding group algebra. We consider an involution on FG which is the linear extension of an involution of G, e.g., g * = g − for g ∈ G. This paper is focused on the characterization of a non-torsion group G provided the group of units U(FG) satisfies a *-group identity. The torsion case was studied in [7], and when * is the classical involution, this problem was solved in the case of symmetric units in [21].
Star group identities on units of group algebras
Contemporary mathematics, 2017
Let G be a group, F a field and FG the corresponding group algebra. We consider an involution on FG which is the linear extension of an involution of G, e.g., g * = g − for g ∈ G. This paper is focused on the characterization of a non-torsion group G provided the group of units U(FG) satisfies a *-group identity. The torsion case was studied in [7], and when * is the classical involution, this problem was solved in the case of symmetric units in [21].
Nil-generated algebras and group algebras whose units satify a Laurent polynomial identity
2021
Let A be an algebra whose group of units U(A) satisfies a Laurent polynomial identity (LPI). We establish conditions on these polynomials in such a way that nil-generated algebras and group algebras with torsion groups over infinite fields in characteristic p > 0 have nonmatrix identities. We also determine, in the context of group algebras with arbitrary LPI for the group of units, the existence of polynomial identities.
Publicacions Matemàtiques, 1992
A bstract TORSION UNITS IN GROUP RINGS VIKAS BIST Let U(RG) be the unit group of the group ring RG. In this paper we study group rings RG whose support elements of every torsion unit are torsion, where R is either the ring of integers 7L or a field K .
Group rings whose torsion units form a subgroup
Proceedings of the American Mathematical Society, 1981
Denote by TU(ZG) the set of units of finite order of the integral group ring of a group G. We determine the class of all groups G such that TUÇLG) is a subgroup and study how this property relates to certain properties of the unit groups.
Symmetric units and group identities
manuscripta mathematica, 1998
In this paper we study rings R with an involution whose symmetric units satisfy a group identity. An important example is given by F G, the group algebra of a group G over a field F; in fact FG has a natural involution induced by setting g + g-1 for all group elements g E G. In case of group algebras if F is infinite, char F 1= 2 and G is a torsion group we give a characterization by proving the following: the symmetric units satisfy a group identity if and only if either the group of units satisfies a group identity (and a characterization is known in this case) or charF = p > 0 and 1) FG satisfies a polynomial identity, 2) the p-elements of G form a (normal) subgroup P of G and G/ P is a Hamiltonian 2-group; 3) G is of bounded exponent 4ps for some s~O.
On group identities for the unit group of algebras and semigroup algebras over an infinite field
Journal of Algebra, 2005
Let k be an infinite field. We fully describe when the unit group of a semigroup algebra k[S] of a semigroup S generated by finitely many periodic elements satisfies a group identity. This and some other recent results are proved by first showing that semiprime k-algebras generated by units are necessarily reduced whenever their unit group satisfies a group identity. 2004 Elsevier Inc. All rights reserved. ✩ Research partially supported by the Onderzoeksraad of Vrije Universiteit Brussel and Fonds voor Wetenschappelijk Onderzoek (Vlaanderen), Flemish-Polish bilateral agreement BIL 01/31, FAPESP-Brazil and CNPq-Brazil (Proc. 300652/95-0).
Structure of group rings and the group of units of integral group rings: an invitation
Indian Journal of Pure and Applied Mathematics, 2021
During the past three decades fundamental progress has been made on constructing large torsion-free subgroups (i.e. subgroups of finite index) of the unit group U (ZG) of the integral group ring ZG of a finite group G. These constructions rely on explicit constructions of units in ZG and proofs of main results make use of the description of the Wedderburn components of the rational group algebra QG. The latter relies on explicit constructions of primitive central idempotents and the rational representations of G. It turns out that the existence of reduced two degree representations play a crucial role. Although the unit group is far from being understood, some structure results on this group have been obtained. In this paper we give a survey of some of the fundamental results and the essential needed techniques.