# Duality in non-abelian algebra IV. Duality for groups and a universal isomorphism theorem

## TL;DR

This paper develops a self-dual axiomatic framework for non-abelian group-like structures, enabling the derivation of fundamental homomorphism theorems and a universal isomorphism theorem, extending classical results beyond abelian categories.

## Contribution

It introduces a novel self-dual context for non-abelian structures, solving a seventy-year-old open problem and unifying isomorphism theorems under a universal framework.

## Key findings

- Established a self-dual axiomatic framework for non-abelian groups.
- Proved a universal isomorphism theorem applicable to all isomorphism theorems.
- Extended classical homomorphism theorems to non-abelian settings.

## Abstract

Abelian categories provide a self-dual axiomatic context for establishing homomorphism theorems (such as the isomorphism theorems and homological diagram lemmas) for abelian groups, and more generally, modules. In this paper we describe a self-dual context which allows one to establish the same theorems in the case of non-abelian group-like structures; the question of whether such a context can be found has been left open for seventy years. We also formulate and prove in our context a universal isomorphism theorem from which all other isomorphism theorems can be deduced.