On integers that are covering numbers of groups

The covering number of a group $G$, denoted by $\sigma(G)$, is the size of a minimal collection of proper subgroups of $G$ whose union is $G$. We investigate which integers are covering numbers of groups. We determine which integers $129$ or smaller are covering numbers, and we determine precisely or bound the covering number of every primitive monolithic group with a degree of primitivity at most $129$ by introducing effective new computational techniques. Furthermore, we prove that, if $\mathscr{F}_1$ is the family of finite groups $G$ such that all proper quotients of $G$ are solvable, then $\mathbb{N}-\{\sigma(G):G\in \mathscr{F}_1\}$ is infinite, which provides further evidence that infinitely many integers are not covering numbers. Finally, we prove that every integer of the form $(q^m-1)/(q-1)$, where $m\neq3$ and $q$ is a prime power, is a covering number, generalizing a result of Cohn.