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arXiv · 2409.20495

Transitivity in wreath products with symmetric groups

Lukas Klawuhn·Kai-Uwe Schmidt
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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 $S_n$.

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Lukas Klawuhn, Kai-Uwe Schmidt. 2024-09-30. Transitivity in wreath products with symmetric groups. https://arxiv.org/abs/2409.20495

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