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

Smallest cyclically covering subspaces of $\mathbb{F}_q^n$, and lower bounds in Isbell's conjecture