On Integers that are Covering Numbers of Groups

The covering number of a group G, denoted by $\sigma \left(G\right),$ 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 $F_1$ is the family of finite groups G such that all proper quotients of G are solvable, then $N-\left\{\sigma \right(G):G\in F_1\}$ is infinite, which provides further evidence that infinitely many integers are not covering numbers. Finally, we prove that every integer of the form $(qm-1)/(q-1),$ where $m\ne 3$ and q is a prime power, is a covering number, generalizing a result of Cohn.