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.     

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 permutation characters of wreath products

It is known that the character rings of symmetric groups S_n and the character rings of hyperoctahedral groups S₂?S_n are generated by (transitive) permutation characters. These results of Young are generalized to wreath products G?H (G a finite group, H a permutation group acting on a finite set). It is shown that the character ring of G?H is generated by permutation characters if this holds for G, H and certain subgroups of H. This result can be sharpened for wreath products G?S_n;if the character ring of G has a basis of transitive permutation characters, then the same holds for the character ring of G?S_n.     

Three types of inclusions of innately transitive permutation groups into wreath products in product action

A permutation group is innately transitive if it has a transitive minimal normal subgroup, and this subgroup is called a plinth. In this paper we study three special types of inclusions of innately transitive permutation groups in wreath products in product action. This is achieved by studying the natural Cartesian decomposition of the underlying set that corresponds to the product action of the wreath product. Previously we identified six classes of Cartesian decompositions that can be acted upon transitively by an innately transitive group with a non-abelian plinth. The inclusions studied in this paper correspond to three of the six classes. We find that in each case the isomorphism type of the acting group is restricted, and some interesting combinatorial structures are left invariant. We also give a fairly general construction of inclusions for each type.     

On the embedding of central extensions into permutation wreath products

We find a necessary condition for embedding a central extension of a group G with elementary abelian kernel into the wreath product that corresponds to a given permutation action of G.     

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     

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.     

Wreath products of finitary permutation groups

We investigate the structure of finitary permutation groups indecomposable into a direct sum.     

On the permutation groups of cyclic codes

We classify the permutation groups of cyclic codes over a finite field. As a special case, we find the permutation groups of non-primitive BCH codes of prime length. In addition, the Sylow p-subgroup of the permutation group is given for many cyclic codes of length p ^m . Several examples are given to illustrate the results.     

The graphical complexity of direct products of permutation groups

In this article, we improve known results, and, with one exceptional case, prove that when k≥3, the direct product of the automorphism groups of graphs whose edges are colored using k colors, is itself the automorphism group of a graph whose edges are colored using k colors. We have handled the case k = 2 in an earlier article. We prove similar results for directed edge‐colored graphs. © 2010 Wiley Periodicals, Inc. J Graph Theory 66:303‐318, 2011     

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.     

Graphical complexity of products of permutation groups

In this paper we study representations of permutation groups as automorphism groups of colored graphs and supergraphs. In particular, we consider how such representations for various products of permutation groups can be obtained from representations of factors and how the degree of complexity increases in such constructions.