Transitivity in wreath products with symmetric groups

TL;DR

This paper investigates transitive subsets within wreath products involving symmetric groups and finite abelian groups, providing structural characterizations, explicit constructions, and connections to orthogonal polynomials and combinatorial designs.

Contribution

It extends the theory of transitive subsets to wreath products, generalizing known results and linking to algebraic combinatorics and coding theory.

Findings

Characterization of transitive subsets using character theory

Generalization of the Livingstone-Wagner theorem

Connections established between transitive sets and Charlier polynomials

Abstract

It is known that the notion of a transitive subgroup of a permutation group $P$ extends naturally to the subsets of $P$. We study transitive subsets of the wreath product $G \wr S_n$, where $G$ is a finite abelian group. This includes the hyperoctahedral group for $G=C_2$. We give structural characterisations of transitive subsets using the character theory of $G \wr S_n$ and interpret such subsets as designs in the conjugacy class association scheme of $G \wr S_n$. In particular, we prove a generalisation of the Livingstone-Wagner theorem and give explicit constructions of transitive sets. Moreover, we establish connections to orthogonal polynomials, namely the Charlier polynomials, and use them to study codes and designs in $C_r \wr S_n$. Many of our results extend results about the symmetric group $S_n$.