Direct products of finite groups as unions of proper subgroups

TL;DR

This paper characterizes all the ways a direct product of two finite groups can be represented as a union of proper subgroups with minimal possible number, revealing structural insights into subgroup unions.

Contribution

It provides a complete classification of direct products of finite groups that can be expressed as unions of proper subgroups with minimal cardinality.

Findings

Identifies all such unions for direct products of finite groups.

Establishes minimal cardinality conditions for subgroup unions.

Provides structural criteria for these unions.

Abstract

We determine all the ways in which a direct product of two finite groups can be expressed as the set-theoretical union of proper subgroups in a family of minimal cardinality.