  For a prime power $q$ and a positive integer $n$, we say a subspace $U$ of
${\mathbb{F}_q^n}$ is {\em cyclically covering} if the union of the cyclic
shifts of $U$ is equal to $\mathbb{F}_q^n$. We investigate the problem of
determining the minimum possible dimension of a cyclically covering subspace of
$\mathbb{F}_q^n$. (This is a natural generalisation of a problem posed in 1991
by the first author.) We prove several upper and lower bounds, and for each
fixed $q$, we answer the question completely for infinitely many values of $n$
(which take the form of certain geometric series). Our results imply lower
bounds for a well-known conjecture of Isbell, and a generalisation theoreof,
supplementing lower bounds due to Spiga. We also consider the analogous problem
for general representations of groups. We use arguments from combinatorics,
representation theory and finite field theory.
