Automorphism group of the variant of the lattice of partitions of a finite set

In this paper we consider variants of the lattice of partitions of a finite set and study automorphism groups of this variants. We obtain irreducible generating sets for of the lattice of partitions of a finite set. We prove that the automorphism group of the variant of the lattice of partitions of a finite set is a natural generalization of the wreath product. The first multiplier of this generalized wreath product is the direct product of the wreaths products, such that depends on the type of the variant generating partition and the second is defined by the certain set of symmetric groups.

CI-Property for Decomposable Schur Rings over an Abelian Group

A Schur ring over a finite group is said to be decomposable if it is the generalized wreath product of Schur rings over smaller groups. In this paper we establish a sufficient condition for a decomposable Schur ring over the direct product of elementary abelian groups to be a CI-Schur ring. By using this condition we offer short proofs for some known results on the CI-property for decomposable Schur rings over an elementary abelian group of rank at most 5.

Separable ultrametric spaces and their isometry groups

The paper is devoted to a study of isometry groups of Polish ultrametric spaces. We explicitly describe isometry groups of spaces that are non-locally rigid and satisfy the property that distances between orbits under the action of the isometry group are realized by points. The type of group construction appearing here is a variant of the generalized wreath product. We prove that it has a natural universality and uniqueness property. As an application, we characterize Polish ultrametric spaces satisfying the above properties, whose isometry groups have uncountable strong cofinality.