Units of compatible nearrings

We study singly-generated wavelet systems on ${\mathbb {R}^2}$ that are naturally associated with rank-one wavelet systems on the Heisenberg group N. We prove a necessary condition on the generator in order that any such system be a Parseval frame. Given a suitable subset I of the dual of N, we give an explicit construction for Parseval frame wavelets that are associated with I. We say that ${g\in L^2(I\times \mathbb {R})}$ is Gabor field over I if, for a.e. ${\lambda \in I}$ , |??|^1/2 g(??, ·) is the Gabor generator of a Parseval frame for ${L^2(\mathbb {R})}$ , and that I is a Heisenberg wavelet set if every Gabor field over I is a Parseval frame (mother-)wavelet for ${L^2(\mathbb {R}^2)}$ . We then show that I is a Heisenberg wavelet set if and only if I is both translation congruent with a subset of the unit interval and dilation congruent with the Shannon set.     

On semidirect products and wreath products of partially ordered groups

The notions of Cartesian and semidirect products for partially ordered groups are considered. A series of results on those products of AO mathcal{A}mathcal{O} -groups and interpolation groups is obtained. Some results concerning wreath products of directed groups are obtained.     

On the isomorphism of wreath products of groups

The well-known Neumann theorem on the isomorphism of standard wreath products is generalized to the wreath products of an arbitrary transitive permutation group and an abstract group. Pedagogical Institute, Vinnitsa. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 6, pp. 671–679, June, 1994     

Units of compatible nearrings,IV

This is the fourth in a sequence of papers originating in a effort to study the units of a compatible nearring $R$ satisfying the descending chain condition on right ideals using a faithful compatible module $G$ of $R$ . A key point in this endeavor involves determining $1 + Ann_R(G/H)$ where $H$ is a direct sum of isomorphic minimal $R$ -ideals where success in doing so gives us not only information about the units of $R$ , but also information about $R$ and $J_2(R)$ . In the previous papers, $1 + Ann_R(G/H)$ has been determined whenever $G/H$ does not contain a minimal factor isomorphic to the minimal summands of $H$ . In this paper we determine $1 + Ann_R(G/H)$ when $G/H$ does contain a minimal factor isomorphic to the minimal summands of $H$ . With the completion of the determination of $1 + Ann_R(G/H)$ in all cases, we illustrate how things work in practice by considering the nearrings generated by the inner automorphisms of a finite dihedral group, special linear group, and general linear group and nearrings of congruence preserving functions on an expanded group.     

Subgroup distortion in wreath products of cyclic groups

We study the effects of subgroup distortion in the wreath products , where A is finitely generated abelian. We show that every finitely generated subgroup of has distortion function equivalent to some polynomial. Moreover, for A infinite, and for any polynomial l^k, there is a 2-generated subgroup of having distortion function equivalent to the given polynomial. Also, a formula for the length of elements in arbitrary wreath product easily shows that the group has distorted subgroups, while the lamplighter group has no distorted (finitely generated) subgroups. In the course of the proof, we introduce a notion of distortion for polynomials. We are able to compute the distortion of any polynomial in one variable over Z,R or C.     

Units of compatible nearrings, III

This paper is a continuation of two previous works studying the units of a compatible nearring R satisfying the descending chain condition on right ideals using a faithful compatible module G of R. A crucial point in doing this involves determining 1 +  Ann _R (G/H) where H is a direct sum of isomorphic minimal R-ideals. The high point of this paper is extending this determination from the cases in the previous works to the case where G/H and H contain no isomorphic minimal factors. We also shall further expand our knowledge of when a special type of principal series for G introduced in the second of these previous works called a quasi c-chain exists.     

Representation theory of wreath products of finite groups

This is an introduction to the representation theory of wreath products of finite groups. We also discuss in full details a couple of examples. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 50, Functional Analysis, 2007.     

Quasivarieties of groups closed under wreath products and wreathZ-products

LetL _q(M) be a lattice of quasivarieties contained in a quasivarietyM. The quasivariety is closed under direct wreath Z-products if together with a group G, it contains its wreath product G ≀ Z with an infinite cyclic group Z. We prove the following: (a) ifM is closed under direct wreath Z-products then every quasivariety, which is a coatom inL _q(M), is likewise closed under these; (b) ifM is closed under direct wreath products thenL _q(M) has at most one coatom. An example of a quasivariety is furnished which is closed under direct wreath Z-products and whose subquasivariety lattice contains exactly one coatom. Also, it turns out that the set of quasivarieties closed under direct wreath Z-products form a complete sublatttice of the lattice of quasivarieties of groups. Supported by RFFR grant No. 96-01-00088, and by the RF Committee of Higher Education. Translated fromAlgebra is Logika, Vol. 38, No. 3, pp. 257–268, May–June, 1999.     

On the representation theory of wreath products of finite groups and symmetric groups

Let G S_N be the wreath product of a finite group G and the symmetric group S_N. The aim of this paper is to prove the branching theorem for the increasing sequence of finite groups G S₁ G S₂ ... G S_N ... and the analog of Young's orthogonal form for this case, using the inductive approach invented by A. Vershik and A. Okounkov for the case of symmetric group.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 240, 1997, pp. 229–244.     

On prime radicals and wreath products of partially ordered groups

The notion of a wreath product for partially ordered groups is considered. A series of results on semidirect products of $ \mathcal{A}\mathcal{O} $ -groups and interpolation groups is obtained. Some results are obtained concerning prime radicals of directed groups.