On the decidability of the integrability of finite groups

TL;DR

This paper proves that determining whether a finite group has an integral, a group whose commutator subgroup is isomorphic to it, is a decidable problem.

Contribution

It establishes the decidability of the integrability problem for finite groups, a question previously unresolved in group theory.

Findings

Integrability of finite groups is decidable.

Provides an algorithmic approach to determine integrability.

Advances understanding of the structure of finite groups.

Abstract

An integral of a group $G$ is a group $H$ whose commutator subgroup is isomorphic to $G$. In this paper, we prove that the integrability of a finite group is a decidable problem.