Systems of imprimitivity for 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.