Every finite group admits a just finite presentation

2026-05-11

AI Summary

This work resolves Problem 21.10 from the *Kourovka Notebook*, which asks whether every finite group admits a "just-finite" presentation—one in which the removal of any single defining relation yields an infinite group. By combining techniques from combinatorial group theory and algebraic constructions, we provide the first affirmative answer, proving that every finite group can be endowed with a minimal presentation whose set of relators is irredundant: omitting any one relator necessarily results in an infinite group. This establishes a universal correspondence between finite groups and their minimally redundant presentations, offering new structural insights into the theory of group presentations.

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.

Problem

• finite group
• just finite presentation
• group presentation
• Kourovka Notebook

Innovation

• just finite presentation
• finite group
• group presentation
• Kourovka Notebook
• combinatorial group theory