Let $G\subset GL_n(q)=GL(V)$, where $V$ is a vector space over a field of $q$ elements. Let $W\subset V$ be a subspace. One says $(G,W)$ is a covering of $V$ if $V$ is the union of $gW: g\in G$. Of my main interest is a special case when $W$ is of dimension $n-1$, $q$ a prime and $G$ is a cyclic group of prime order acting on $V$ irreducibly. Not much is known in this situation, there are more questions than results.