# Coproducts of Finite Groups

## TL;DR

This paper proves that the coproduct of any two non-trivial finite groups does not exist within the category of finite groups, highlighting a fundamental limitation in the structure of finite groups.

## Contribution

It establishes a non-existence result for coproducts in the category of finite groups, providing new insights into their categorical properties.

## Key findings

- Coproducts of non-trivial finite groups are not representable.
- The result clarifies limitations in categorical constructions within finite group theory.
- Highlights the difference between finite and infinite group categories.

## Abstract

We show that for any pair of non-trivial finite groups, their coproduct in the category of finite groups is not representable.