Systems of imprimitivity for wreath products

TL;DR

This paper investigates the structure of systems of imprimitivity in certain linear groups, especially wreath products, establishing conditions for their uniqueness and exploring implications for subgroup inclusions.

Contribution

It characterizes when nonrefinable systems of imprimitivity are unique in wreath product groups, revealing a specific exceptional case and applying findings to subgroup inclusion problems.

Findings

Unique nonrefinable system of imprimitivity in most cases

Exceptional case where uniqueness fails: d=1, n=k even, |H|=2

Results on subgroup inclusions in wreath products

Abstract

Let $G$ be an irreducible imprimitive subgroup of $\operatorname{GL}_n(\mathbb{F})$, where $\mathbb{F}$ is a field. Any system of imprimitivity for $G$ can be refined to a nonrefinable system of imprimitivity, and we consider the question of when such a refinement is unique. Examples show that $G$ can have many nonrefinable systems of imprimitivity, and even the number of components is not uniquely determined. We consider the case where $G$ is the wreath product of an irreducible primitive $H \leq \operatorname{GL}_d(\mathbb{F})$ and transitive $K \leq S_k$, where $n = dk$. We show that $G$ has a unique nonrefinable system of imprimitivity, except in the following special case: $d = 1$, $n = k$ is even, $|H| = 2$, and $K$ is a transitive subgroup of $C_2 \wr S_{n/2}$. As a simple application, we prove results about inclusions between wreath product subgroups.