Equal Coverings of Finite Groups

TL;DR

This paper investigates the conditions under which finite groups have equal coverings, developing theorems and utilizing computational tools to determine their existence.

Contribution

It introduces new theorems for identifying equal coverings in finite groups and combines theoretical and computational methods for analysis.

Findings

Development of theorems for equal coverings

Application of GAP for computational verification

Insights into which finite groups admit equal coverings

Abstract

We will explore the nature of when certain finite groups have an equal covering, and when finite groups do not. Not to be confused with the concept of a cover group, a covering of a group is a collection of proper subgroups whose set-theoretic union is the original group. We will discuss the history of what has been researched in the topic of coverings, and as well as mention some findings in concepts related to equal coverings such as that of equal partition of a group. We develop some useful theorems that will aid us in determining whether a finite group has an equal covering or not. In addition, for when a theorem may not be entirely useful to examine a certain group we will turn to using \texttt{\texttt{GAP}} (Groups, Algorithms, Programming) for computational arguments.