The finite images of finitely generated groups

Given any sequence of non-abelian finite simple primitive permutationgroups S_n, we construct a finitely generated group G whose profinitecompletion is the infinite permutational wreath product ...S_n S_n–1 ... S₀. It follows that the upper compositionfactors of G are exactly the groups S_n. By suitably choosingthe sequence S_n we can arrange that G has any one of a continuousrange of slow, non-polynomial subgroup growth types. We alsoconstruct a 61-generator perfect group that has every non-abelianfinite simple group as a quotient. 2000 Mathematics SubjectClassification: 20E07, 20E08, 20E18, 20E32.     

On an identity of Ramanujan over finitely generated domains

Summary We show that a well-known identity of Ramanujan admits only a bounded number of solutions over general finitely generated domains. The bound is explicit and uniform in the sense that it depends only on the dimensions of the domains involved. Our method is constructive, and opens up a possibility to determine the solutions in concrete instances. In some special cases all solutions are determined. Our results can also be considered as a continuation of some theorems of Z. Daróczy and G. Hajdu, obtained over Z. We note that in case of Hosszú's equation, similar results were obtained by several authors.     

On monomial representations of finitely generated nilpotent groups

A result of Segal states that every complex irreducible representation of a finitely generated nilpotent group G is monomial if and only if G is abelian-by-finite. A conjecture of Parshin, recently proved affirmatively by Beloshapka and Gorchinskii (2016), characterizes the monomial irreducible representations of finitely generated nilpotent groups. This article gives a slightly shorter proof of the conjecture using ideas of Kutzko and Brown. We also give a characterization of the finite-dimensional irreducible representations of two-step nilpotent groups and describe these completely for two-step groups whose center has rank one.     

Quasi-Whittaker modules over the conformal Galilei algebras

In this paper, we study quasi-Whittaker modules over the conformal Galilei algebras. We determine all quasi-Whittaker vectors and the simplicities of the universal quasi-Whittaker modules. For the reducible ones, we determine the corresponding simple quotient modules. Thus we classify all simple quasi-Whittaker modules for the conformal Galilei algebras.     

Existence of critical modules of GK-dimension 2 over elliptic algebras

We show that over an elliptic algebra, critical modules of Gelfand-Kirillov dimension 2 exist in all multiplicities (assuming the ground field is uncountable, algebraically closed). Geometrically, this shows that in a quantum plane there exist ``irreducible curve" modules of all possible degrees.

     

Varieties and cohomology of infinitely generated modules

Suppose that k is a field of characteristic p and that G is a finite group having p-rank at least three. Given a regular sequence of length 2 in H*(G, k), we show how to construct an infinitely generated module whose support variety is strictly smaller than the small support of its cohomology. Received: 25 January 2008     

On finitely generated multiplication modules

We shall prove that if M is a finitely generated multiplication module and Ann(M) is a finitely generated ideal of R, then there exists a distributive lattice M such that Spec(M) with Zariski topology is homeomorphic to Spec(M) to Stone topology. Finally we shall give a characterization of finitely generated multiplication R-modules M such that Ann(M) is a finitely generated ideal of R.     

Two finitely generated varieties having no infinite simple members

Using a method of R. McKenzie, we construct a finitely generated semisimple variety of infinite type, and a finitely generated nonsemisimple variety of finite type, both having arbitrarily large finite but no infinite simple members. This amplifies M. Valeriote's negative solution to Problem 11 from Hobby and McKenzie, The Structure of Finite Algebras.

     

Periodic modules over general quantum Laurent polynomials

We construct, by induction on the number of variables, periodic finitely generated modules over general quantum Laurent polynomials. Translated fromMatematicheskie Zametki, Vol. 61, No. 1, pp. 10–17, January, 1997. Translated by A. I. Shtern     

Construction of free subgroups in the group of units of modular group algebras

Let KGbe the group algebra of a p¹ -group Gover a field Kof characteristic p > 0, and let U(KG)be its group of units. If KGcontains a nontrivial bicyclic unit and if Kis not algebraic over its prime field, then we prove that the free product Z_p? Z_p? Z_pcan be embedded in U(KG).     

The Brauer-Clifford group for locally finite (S,H)-Azumaya algebras

In this article, we define the notion of Brauer-Clifford group for H-locally finite (S, H)-Azumaya algebras, when H is a cocommutative Hopf algebra and S is an H-locally finite commutative H-module algebra over a commutative noetherian ring k. This is the situation that arises in applications with connections to algebraic geometry. This Brauer-Clifford group turns out to be an example of a Brauer group of a symmetric monoidal category.