On certain groups with $G$-contrainvariant subspaces

Abstract

For the field $\mathbb{𝔽}$ and $\mathbb{𝔽}$-vector space $A$, let $G$ be a subgroup of $GL\left(\mathbb{𝔽},A\right)$, the group of invertible $\mathbb{𝔽}$-linear transformations of $A$. A subspace $B$ of $A$ is $G$-contrainvariant if $GB=A$. We discuss groups $G$ and vector spaces $A$ such that every subspace of $A$ is either $G$-invariant or $G$-contrainvariant.