Symmetric units and group identities

In this paper we study rings R with an involution whose symmetric units satisfy a group identity. An important example is given by FG, 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∈G. In case of group algebras if F is infinite, charF≠ 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 char F=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 4p ^s for some s≥ 0. Received: 8 August 1997     

Twisted identities in Coxeter groups

Given a Coxeter system (W,S) equipped with an involutive automorphism θ, the set of twisted identities is

Conditional identities in finite groups

Siberian Mathematical Journal -     

Menon-type identities with respect to sets of units

The Ramanujan Journal - Let $$n\ge 1$$ and let $$\mathbb {Z}_n^\star $$ be the group of units in the ring of residual classes modulo n, let $$m\ge 0$$ , $$k\ge 0$$ , $$m+k\ge 1$$ , $$u_1,\ldots...     

Two-variable identities for finite solvable groups

We characterise the solvable groups in the class of finite groups by an inductively defined sequence of two-variable identities. Our main theorem is the analogue of a classical theorem of Zorn which gives a characterisation of the nilpotent groups in the class of finite groups by a sequence of two-variable identities. To cite this article: T. Bandman et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Generalized group identities in classical groups

A detailed exposition of results announced in RZhMat 1981, 4A186, concerning the nonexistence of generalized group identities in classical groups over a division ring with infinite center is shown.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 96–119, 1982.The authors would like to express their gratitude to A. I. Kostrikin for posing this problem and to V. N. Latyshev for his interest and useful comments.     

Polynomial identities and maximal subgroups of skew linear groups

Suppose that D is a division ring with center F and N is a non-central normal subgroup of GL _n (D). In this paper we generalize some known results about maximal subgroups of GL _n (D) to maximal subgroups of N. More precisely, we prove that if M is a maximal subgroup of N such that F[M] satisfies a polynomial identity and contains an algebraic element over F or and either n ≥ 2 or n = 1 and M is not abelian, then [D : F] < ∞. This research was partially supported by a grant from IPM (No. 85160047).     

Nil-extensions of unions of groups induced by identities

This paper is a continuation of [11]. Here we study all identities which induce decompositions into nil-extensions of a union of groups and identities which induce various special types of these decompositions. Supported by Grant 0401A of SF Servia through Math. Inst. SANU.     

On identities in groups of fractions of cancellative semigroups

To solve two problems of Bergman stated in 1981, we construct a group such that contains a free noncyclic subgroup (hence, satisfies no group identity) and , as a group, is generated by its subsemigroup that satisfies a nontrivial semigroup identity.

     

Group identities on units of locally finite algebras and twisted group algebras

We study locally finite algebras and twisted group algebras with units satisfying a group identity. As a preliminary result, we obtain a necessary condition for twisted group algebras to satisfy a generalized polynomial identity.     

Bases of the identities of products of certain varieties of groups

The following theorem is proved: The product of any variety of two-step solvable groups and a variety having a finite basis of identity relations has a finite basis of identity relations.Translated from Matematicheskie Zametki, Vol. 5, No. 1, pp. 137–144, January, 1969.     

Group algebras of Kleinian type and groups of units

The algebras of Kleinian type are finite-dimensional semisimple rational algebras A such that the group of units of an order in A is commensurable with a direct product of Kleinian groups. We classify the Schur algebras of Kleinian type and the group algebras of Kleinian type. As an application, we characterize the group rings RG, with R an order in a number field and G a finite group, such that the group of units of RG is virtually a direct product of free-by-free groups.