Transitivity in wreath products with symmetric groups

2024/09/30 by Lukas Klawuhn, Klawuhn, Lukas, Kai‐Uwe Schmidt +1
Mathematics · #05B99 #05E30 #20C99 #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #Mathematics and Applications #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2409.20495

openalex publication_date 2024/09/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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=C₂. 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.

Related

• Transitive simple subgroups of wreath products in product action
• A new notion of transitivity for groups and sets of permutations
• A Relationship Between Character Values Of Wreath Products And The Symmetric Group
• Transitivity in finite general linear groups
• Innately transitive subgroups of wreath products in product action