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Every finite group admits a just finite presentation

The paper resolves the open question from the Kourovka Notebook (Problem 21.10) by proving that every finite group admits a "just finite" presentation, meaning a presentation where the removal of any single relation results in an infinite group.

Original authors: Marc Lackenby

Published 2026-05-12

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Marc Lackenby

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a group of friends trying to solve a puzzle. In the world of mathematics, these "friends" are called finite groups. They are collections of rules and moves that, if you keep playing them, eventually loop back to the start. They are small, manageable, and predictable.

For a long time, mathematicians wondered if there was a special way to write down the rules for any of these groups. They wanted a set of instructions where every single rule was absolutely essential.

The "Just Finite" Puzzle

Think of a group's presentation as a recipe.

The Ingredients (Generators): The basic moves you can make (like "turn left" or "jump").
The Rules (Relations): The instructions that tell you when a sequence of moves brings you back to the starting point (like "if you turn left four times, you are back where you started").

Usually, a recipe might have a few extra rules that aren't strictly necessary. If you remove one, the dish still tastes the same. But a "Just Finite" presentation is a recipe where every single rule is critical.

If you take away even one rule from this special recipe, the result is chaos. The group stops being a small, finite circle of friends and explodes into an infinite crowd. The rules that used to keep everyone in check are gone, and the group runs off to infinity.

For decades, mathematicians asked: Can we always find such a "perfect" recipe for any finite group? This was a famous open question known as Problem 21.10 in a notebook called the Kourovka Notebook.

The Solution: The "Double-Trap" Trick

The author of this paper, Marc Lackenby (with significant help from an AI co-mathematician), says yes. Every finite group has such a presentation.

Here is the clever trick they used, explained simply:

Imagine you have a rule in your recipe that says, "If you do X, you are back at the start."

The Problem: If you just remove this rule, maybe the group stays finite, or maybe it becomes infinite. You can't guarantee it becomes infinite.
The Fix: Instead of keeping the rule as is, the author replaces it with a two-part trap.
- They introduce a new, dummy character (let's call him "Bob").
- They create two new rules involving Bob and the original rule.
- Rule A: "If you do the original move, Bob changes in a specific way."
- Rule B: "If Bob does the original move, the original move changes in a specific way."

These two rules are designed like a mathematical lock. As long as both rules are present, they force "Bob" to be nothing (he disappears) and the original rule to be true. The group remains exactly the same size.

But here is the magic:

If you remove Rule A, the lock breaks. The group suddenly gains the ability to stretch out forever (it becomes infinite).
If you remove Rule B, the lock breaks in a different way, and the group also stretches out forever.

By swapping every single original rule for this "double-trap" pair, the author ensures that no matter which rule you remove, the group explodes into infinity.

Why This Matters (In Math Terms)

The paper proves that for any finite group, you can construct a presentation where removing any relation destroys the "finite" nature of the group.

The author also showed this works for groups with a special property called Property (FA) (which means the group can't be split apart easily) and Property (T) (a very rigid type of group). Since all finite groups have Property (FA), the main result holds true for them all.

A Note on the "Co-Author"

The paper has a unique twist in its "Methodology" section. The author used an AI tool (Google DeepMind's AI co-mathematician) to help solve this problem.

The AI came up with the core idea and the "double-trap" construction.
However, the AI hit a snag: it wasn't sure what to do if a specific part of the group was just a simple circle (a cyclic group).
The human author stepped in, analyzed the gap, and found the missing piece of logic to fix the proof.
The AI then confirmed the fix was correct and helped write the final paper.

Summary

In short, the paper solves a decades-old puzzle by showing that every finite group can be described by a set of rules where every single rule is the only thing keeping the group from running away to infinity. It's like building a cage where every bar is essential; remove just one, and the animal escapes.

Technical Summary: Every Finite Group Admits a Just Finite Presentation

Problem Statement
The paper addresses Problem 21.10 from the Kourovka Notebook, posed by Barnea. The problem asks whether every finite group admits a "just finite" presentation. A finite presentation $\langle X \mid R \rangle$ of a finite group $G$ is defined as just finite if the removal of any single relation $r \in R$ results in a presentation for an infinite group. While the theory of "just infinite" groups (infinite groups where every proper quotient is finite) is well-established, the dual concept for finite groups remained an open question.

Methodology and Construction
The author resolves the conjecture by proving a stronger result concerning groups with Serre's Property (FA). A group has Property (FA) if every action on a tree has a fixed point; for finitely generated groups, this is equivalent to the group being neither a non-trivial amalgamated free product nor an HNN extension. Since all finite groups possess Property (FA), proving the existence of a "just-(FA)" presentation for any finitely presented group with Property (FA) suffices to solve the original problem.

The core construction modifies an existing irredundant finite presentation $\langle X \mid R \rangle$ of a group $G$ (where $G$ is non-cyclic) into a new presentation $P' = \langle X' \mid R' \rangle$ :

Generator Expansion: For every relation $r \in R$ , a new generator $b_r$ is introduced.
Relation Substitution: Each original relation $r$ $r$ is replaced by a pair of new relations:
- $r^{-1} b_r r = b_r^2$
- $b_r^{-1} r b_r = r^2$
Theoretical Basis: This substitution relies on Lemma 4 (B.H. Neumann), which states that if elements $u, v$ satisfy $u^{-1}vu = v^2$ and $v^{-1}uv = u^2$ , then $u=v=1$ . Consequently, in the new presentation $P'$ , the relations force $b_r = 1$ and $r = 1$ for all $r$ , ensuring that $P'$ still presents the original group $G$ .

Proof of Just-(FA) Property
To demonstrate that $P'$ is just-(FA), the author shows that removing any single relation from $R'$ yields a group that lacks Property (FA). The proof analyzes two cases for a specific relation pair corresponding to an original relation $r$ :

Case 1 (Removing $b_r^{-1} r b_r = r^2$ ): The resulting group $K_1$ is shown to be a non-trivial amalgamated free product. Specifically, it decomposes as $H_r *_{\langle r \rangle} B'$ , where $H_r = \langle X \mid R \setminus \{r\} \rangle$ and $B'$ is a semi-direct product (or a Baumslag-Solitar group if the order of $r$ is infinite). The index of the amalgamating subgroup is shown to be at least 3, ensuring the product is non-trivial unless $H_r$ is cyclic. The author argues that if $H_r$ were cyclic, $G$ (as a quotient of $H_r$ ) would also be cyclic, contradicting the assumption that $G$ is non-cyclic. Thus, $K_1$ is a non-trivial amalgamated free product and lacks Property (FA).
Case 2 (Removing $r^{-1} b_r r = b_r^2$ ): The resulting group $K_2$ admits a surjective homomorphism onto the infinite cyclic group $\mathbb{Z}$ . By mapping all original generators to the identity and the specific generator $b_r$ to a generator of $\mathbb{Z}$ , the remaining relation is satisfied trivially. Since $K_2$ maps onto an infinite group, it cannot have Property (FA).

Key Results
The paper establishes the following theorems:

Theorem 1: Every finite group admits a finite, just finite presentation.
Theorem 2: Every finitely presented group with Property (FA) admits a finite just-(FA) presentation.
Theorem 3: Every finitely presented group with Kazhdan's Property (T) admits a finite just-(T) presentation (a presentation where removing any relation destroys Property (T)).
Theorem 6: For any finite irredundant presentation of a non-cyclic group with Property (FA), there exists a just-(FA) presentation with the same deficiency (number of generators minus number of relations).

Significance and Context
The paper claims to resolve the open question posed in the Kourovka Notebook in the affirmative. The significance lies in establishing the existence of minimal presentations for finite groups in the sense that no relation is redundant for maintaining finiteness.

Methodology Note
The paper explicitly acknowledges the use of the Google DeepMind AI co-mathematician tool. The AI proposed the initial solution structure and the core proof strategy but encountered a flaw regarding the specific case where $H_r = \langle r \rangle$ in Case 1. The human author identified the logical gap, provided the necessary fix (the argument regarding cyclic quotients), and refined the presentation to include the broader concepts of just-(FA) and just-(T) presentations. The final text was verified and corrected by the AI tool.

The "Just Finite" Puzzle

The Solution: The "Double-Trap" Trick

Why This Matters (In Math Terms)

A Note on the "Co-Author"

Summary

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