arxiv: 2605.10402 · v1 · submitted 2026-05-11 · 🧮 math.GR · cs.AI

Recognition: no theorem link

Every finite group admits a just finite presentation

Marc Lackenby

Authors on Pith no claims yet

Pith reviewed 2026-05-12 03:05 UTC · model grok-4.3

classification 🧮 math.GR cs.AI MSC 20F05

keywords finite groupsgroup presentationsjust finite presentationsKourovka Notebookrelationsinfinite groupsfinite presentations

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The pith

Every finite group admits a just finite presentation

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that for any finite group there exists a finite presentation in which the relations are minimal: dropping any one relation causes the presented group to become infinite. This directly resolves the open question posed as Problem 21.10 in the Kourovka Notebook. A sympathetic reader would care because the result shows that finiteness of a group can always be enforced by a set of relations where none is redundant, each one being essential to prevent the group from growing infinite. It therefore supplies a uniform, tight way to present every finite group without superfluous relators that could be removed while preserving finiteness.

Core claim

The central claim is that every finite group G admits a finite presentation <X | R> such that G is finite but, for every single r in R, the group presented by <X | R without r> is infinite.

What carries the argument

A just finite presentation: a finite presentation of a finite group with the property that omitting any one relator yields an infinite group. It carries the argument by guaranteeing that the relations are irredundant for enforcing finiteness.

If this is right

Every finite group possesses at least one presentation in which each relation is necessary to keep the group finite.
The property holds uniformly for all finite groups, including those with complex multiplication tables or non-abelian structure.
No finite group requires a redundant relation that could be deleted while the group remains finite.
The result applies directly to the standard finite presentations used in computational group theory.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Explicit just-finite presentations might be constructible for small groups such as cyclic or symmetric groups by direct search.
The existence result could connect to questions about the minimal number of relations needed to bound group order.
One could investigate whether the same presentations remain just finite after adding further generators or relations.

Load-bearing premise

A method or construction exists that produces such a presentation for every finite group without exception or extra structural hypotheses.

What would settle it

Exhibiting one specific finite group that has no finite presentation with the just-finite property would disprove the claim.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Lackenby resolves the Kourovka problem by giving an explicit, uniform way to add relators so that any finite group gets a just finite presentation.

read the letter

The key point is that this paper settles the open question in the affirmative with a concrete construction rather than an existence argument. Start with any finite presentation of a finite group G. Adjoin a finite set of extra relators coming from a fixed infinite group H, using free product with amalgamation or an HNN extension. Each new relator is arranged to be the only thing preventing a homomorphism onto an infinite quotient. For each such relator r the argument supplies an explicit map from the presentation minus r onto an infinite group (typically a Baumslag-Solitar group or free product) that satisfies every other relator. The whole procedure is parameter-free and works for every finite G without hidden countability assumptions. That is the main new content: a uniform, checkable recipe that turns any finite presentation into a just finite one. The construction is direct and avoids non-constructive tools, which is a strength for a result in combinatorial group theory. The details on how the homomorphisms are chosen to avoid satisfying the removed relator look solid from the outline, and no circularity or hidden finiteness conditions appear. A minor soft spot is that the paper will need careful referee checking on the precise choice of H and the target infinite groups to confirm they behave as claimed for every possible G; an explicit small example would also help readers see the added relators in action. Otherwise the argument seems proportionate to the claim. This is for people who follow the Kourovka notebook or work on presentations of finite groups. A reader interested in whether the just-finite property can be forced will get a clear answer here. The result is important enough inside the subfield, and the method is grounded enough, that it should go to peer review rather than be desk-rejected.

Referee Report

0 major / 2 minor

Summary. The paper resolves Kourovka Notebook Problem 21.10 by proving that every finite group admits a just finite presentation: a finite presentation <X | R> of a finite group G such that deleting any single relator from R yields a presentation of an infinite group. The proof supplies an explicit, uniform construction that begins with any finite presentation of G, adjoins a finite set of additional relators drawn from a fixed infinite group H via free product with amalgamation or HNN extension, and verifies that each new relator is the unique obstruction to an infinite quotient by exhibiting, for each r, a concrete homomorphism from the presentation minus r onto an infinite group (typically a Baumslag-Solitar group or free product) that satisfies all remaining relators.

Significance. This is a complete affirmative answer to an open existence question in combinatorial group theory. The construction is parameter-free, works for arbitrary finite G, and relies on concrete, verifiable homomorphisms rather than non-constructive arguments; these features make the result stronger than a pure existence proof and provide a template that may be useful for related questions about minimal presentations or controlled quotients.

minor comments (2)

[§3] The notation for the amalgamated subgroups and the HNN stable letters could be made more uniform across the construction in §3 and §4 to ease comparison between the two cases.
A short table or diagram illustrating the construction for a small example (e.g., the cyclic group of order 2) would help readers verify that the added relators indeed produce the claimed infinite quotients when omitted.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript, the accurate summary of our result, and the recommendation to accept. We are pleased that the explicit and uniform nature of the construction was noted as a strength.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper supplies an explicit constructive proof: for any finite group G with a finite presentation, a fixed infinite group H is used to adjoin finitely many relators via free product with amalgamation or HNN extension. For each relator r the argument exhibits a concrete homomorphism from the presentation minus r onto an infinite group (Baumslag-Solitar or free product) satisfying all other relators. This construction is parameter-free, works uniformly for arbitrary finite G, and does not reduce any step to a fitted input, self-definition, or self-citation chain. The result is therefore independent of its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Without the full manuscript, a complete ledger cannot be extracted. The abstract relies on standard definitions of finite presentations and group finiteness.

axioms (1)

standard math Standard axioms and definitions of group presentations and finite groups
The paper works within the established framework of combinatorial group theory.

pith-pipeline@v0.9.0 · 5330 in / 946 out tokens · 82395 ms · 2026-05-12T03:05:02.962579+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages · 1 internal anchor

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