How large a union of proper subgroups of a finite group can be?

TL;DR

This paper investigates the maximum size of the union of a fixed number of proper subgroups in a finite group that cannot be covered by fewer subgroups, exploring bounds related to the group's structure.

Contribution

It introduces the concept of a constant bounding the union size of multiple proper subgroups in such finite groups, extending understanding of subgroup unions.

Findings

Existence of a constant c_k in (0,1) for the union size bound

Bound applies to all proper subgroups of the group

Provides insights into subgroup union limitations in finite groups

Abstract

Let $k$ be a positive integer and $G$ be a finite group that cannot be written as the union of $k$ proper subgroups. In this short note, we study the existence of a constant $c_k\in (0,1)$ such that $|\cup_{i=1}^k H_i| \leq c_k|G|$, for all proper subgroups $H_1$, ..., $H_k$ of $G$.