# Isomorphism Theorems for Groupoids and Some Applications

## TL;DR

This paper introduces fundamental properties of groupoids, proves isomorphism theorems, and applies these results to extend classical group theory theorems and relate partial group actions to inverse semigroup actions.

## Contribution

It presents the first comprehensive algebraic treatment of groupoid isomorphism theorems and extends classical theorems like Zassenhaus and Jordan-Hölder to groupoids.

## Key findings

- Established fundamental properties of groupoids.
- Proved isomorphism theorems for groupoids.
- Extended classical group theorems to the groupoid context.

## Abstract

Using an algebraic point of view we present an introduction to the groupoid theory, that is, we give fundamental properties of groupoids as, uniqueness of inverses and properties of the identities, and study subgroupoids, wide subgroupoids and normal subgroupoids. We also present the isomorphism theorems for groupoids and as an applications we obtain the corresponding version of Zassenhaus Lemma and Jordan-H\"{o}lder Theorem for groupoids. Finally inspired by the Ehresmann-Schein-Nambooripad Theorem we improve a result of R. Exel concerning a one to one correspondence between partial actions of groups and actions of inverse semigroups.