Every finite group admits a just finite presentation

TL;DR

This paper proves that every finite group has a finite presentation where removing any relation makes the group infinite, resolving a long-standing open problem.

Contribution

It establishes that all finite groups admit a just finite presentation, confirming a conjecture from the Kourovka Notebook.

Findings

Every finite group admits a just finite presentation.

Removing any relation from the presentation yields an infinite group.

The conjecture has been open prior to this proof.

Abstract

A finite presentation < X | R > of a finite group is called `just finite' if removing any relation from R results in a presentation for an infinite group. It has been an open question (Kourovka Notebook, Problem 21.10) whether every finite group admits such a presentation. We resolve this conjecture in the affirmative.