On integers that are covering numbers of groups

TL;DR

This paper characterizes which integers are covering numbers of groups, determines these numbers for groups up to degree 129, and introduces new computational methods to analyze primitive monolithic groups.

Contribution

It provides a complete classification of covering numbers up to 129, introduces effective computational techniques, and proves new results about the set of integers that are or are not covering numbers.

Findings

All integers up to 129 are classified as covering or not.

Introduces new computational methods for analyzing primitive monolithic groups.

Proves that all integers of the form (q^m-1)/(q-1), with specified conditions, are covering numbers.

Abstract

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.