Referee A: accept / high. Referee B: accept / high. Full consensus on acceptance; both identify only typographic issues in the title and abstract.
The dihedral group of order 8 admits an explicit just-finite presentation obtained by the construction.
The example verifies that the new relations are irredundant.
| ID | Claim | Section | Importance | Status | Lean match | Author action |
|---|---|---|---|---|---|---|
| C6 | The dihedral group of order 8 admits an explicit just-finite presentation obtained by the construction. | Example | clarify before publication | verified | The example verifies that the new relations are irredundant. | |
| C1 | Every finite group admits a finite just-finite presentation. | Theorem 1 | no action needed | verified | Follows immediately from Theorem 2 and the fact that finite groups have Property (FA) by Serre's theorem. | |
| C2 | Every finitely presented group with Property (FA) admits a finite just-(FA) presentation. | Theorem 2 | no action needed | verified | Proved by the augmentation construction using Neumann's lemma. | |
| C3 | Augmenting an irredundant presentation with one new generator b_r per relation and replacing each r by the pair r^{-1} b_r r = b_r^2 and b_r^{-1} r b_r = r^2 still defines the original group G. | Proof of Theorem 2 | no action needed | verified | Direct application of B. H. Neumann's lemma on the given conjugation relations. | |
| C4 | Removing either new relation from the augmented presentation produces a group that lacks Property (FA). | Proof of Theorem 2 | no action needed | verified | The resulting group is either a nontrivial amalgamated free product or admits a surjection onto Z. | |
| C5 | The construction yields just-(T) presentations for groups with Kazhdan's Property (T). | Section on Property (T) | no action needed | verified | Immediate corollary since (T) implies (FA) by Watatani's theorem. | |
| C7 | The construction preserves the deficiency of the original presentation for non-cyclic groups. | Theorem 6 | no action needed | verified | Additional statement that strengthens the result. |
V0 Prose only; no formal artifacts supplied.
The paper resolves Kourovka Notebook Problem 21.10 by proving that every finite group admits a finite just-finite presentation. It establishes the stronger Theorem 2: every finitely presented group with Serre's Property (FA) admits a finite just-(FA) presentation. The proof augments an irredundant presentation with new generators and relations whose removal destroys Property (FA). An explicit example for the dihedral group of order 8 is included, and consequences for groups with Property (T) are noted.
The result settles a named open problem from the Kourovka Notebook with a short, constructive proof. It supplies explicit just-finite presentations for any finite group and preserves deficiency in the non-cyclic case. The argument is self-contained and may have further applications in geometric group theory.
The dihedral group of order 8 admits an explicit just-finite presentation obtained by the construction.
The example verifies that the new relations are irredundant.
Every finite group admits a finite just-finite presentation.
Follows immediately from Theorem 2 and the fact that finite groups have Property (FA) by Serre's theorem.
Every finitely presented group with Property (FA) admits a finite just-(FA) presentation.
Proved by the augmentation construction using Neumann's lemma.
Augmenting an irredundant presentation with one new generator b_r per relation and replacing each r by the pair r^{-1} b_r r = b_r^2 and b_r^{-1} r b_r = r^2 still defines the original group G.
Direct application of B. H. Neumann's lemma on the given conjugation relations.
Removing either new relation from the augmented presentation produces a group that lacks Property (FA).
The resulting group is either a nontrivial amalgamated free product or admits a surjection onto Z.
The construction yields just-(T) presentations for groups with Kazhdan's Property (T).
Immediate corollary since (T) implies (FA) by Watatani's theorem.
The construction preserves the deficiency of the original presentation for non-cyclic groups.
Additional statement that strengthens the result.
Publication readiness is governed by the referee recommendation, required revisions, and the blockers summarized above.
Minor differences in emphasis on proof terseness and AI disclosure.
Notes that the argument that H_r cannot be cyclic is terse and suggests a reminder sentence; highlights transparent AI disclosure as a strength.
Finds the proof self-contained with no need for extra sentences; treats AI disclosure as appropriate but not requiring comment.
The differences are stylistic only; both referees agree the proof is correct and recommend acceptance. The suggested sentence is optional.
Referee A: accept / high. Referee B: accept / high. Full consensus on acceptance; both identify only typographic issues in the title and abstract.
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"canon_match_strength": "none",
"cited_canon_theorems": [],
"confidence": "high",
"issue_inventory": [],
"load_bearing_issues": [],
"major_comments": [],
"minor_comments": [
{
"comment": "The title and abstract contain typographic artifacts (spaces inside \u0027PRESENT A TION\u0027, inconsistent hyphenation of \u0027just finite\u0027 versus \u0027just-finite\u0027). These do not affect readability but should be cleaned before publication.",
"section": "Abstract"
},
{
"comment": "The argument that H_r cannot be cyclic when G is non-cyclic is correct but terse. A single sentence reminding the reader that any quotient of a cyclic group is cyclic would make the contradiction immediate.",
"section": "Proof of Theorem 2, Case 1"
}
],
"optional_revisions": [],
"paper_summary": "The paper resolves Kourovka Notebook Problem 21.10 by proving that every finite group admits a finite just-finite presentation. It establishes the stronger Theorem 2: every finitely presented group with Serre\u0027s Property (FA) admits a finite just-(FA) presentation. The proof begins with an irredundant finite presentation of such a group G, augments the generating set with one new generator b_r per relation r, and replaces each original relation r with the pair of relations r^{-1} b_r r = b_r^2 and b_r^{-1} r b_r = r^2. Neumann\u0027s lemma ensures these force b_r = 1 and r = 1, so the new presentation still defines G. Removing either new relation produces either a non-trivial amalgamated free product (hence no Property (FA)) or a group surjecting onto Z (hence no Property (FA)). Special cases for cyclic groups and extensions to Property (T) are handled directly.",
"recommendation": "accept",
"required_revisions": [],
"significance": "The result settles a 2026 Kourovka Notebook question posed by Barnea. It supplies an explicit, uniform construction that preserves deficiency for non-cyclic groups and yields concrete just-finite presentations. The argument relies only on classical tools (Serre\u0027s theorem, Neumann\u0027s lemma, Baumslag-Solitar groups) and produces falsifiable, checkable presentations for any concrete finite group.",
"strengths": [
"Resolves an open Kourovka Notebook problem with a short, self-contained proof.",
"The construction is uniform and preserves the deficiency of the original presentation.",
"Explicit verification for the dihedral group of order 8 supplies a concrete example.",
"The extension to Property (T) follows immediately from the known implication (T) implies (FA).",
"The AI-assisted methodology is disclosed transparently without affecting the mathematical content."
]
}
{
"canon_match_strength": "none",
"cited_canon_theorems": [],
"confidence": "high",
"issue_inventory": [],
"load_bearing_issues": [],
"major_comments": [],
"minor_comments": [
{
"comment": "The title contains extraneous spaces (\u0027PRESENT A TION\u0027); correct to \u0027PRESENTATION\u0027 for the published version.",
"section": "Title and abstract"
},
{
"comment": "The detailed account of AI co-mathematician assistance is transparent and appropriate; no changes required.",
"section": "Methodology"
}
],
"optional_revisions": [],
"paper_summary": "The paper resolves Kourovka Notebook Problem 21.10 by proving that every finite group admits a finite just finite presentation. It establishes the stronger Theorem 2: every finitely presented group with Serre\u0027s Property (FA) admits a finite just-(FA) presentation. The construction starts from an irredundant finite presentation and replaces each relation r with two new relations involving a fresh generator b_r, using B.H. Neumann\u0027s lemma on elements satisfying u^{-1}vu = v^2 and v^{-1}uv = u^2. Removing either new relation produces either a non-trivial amalgamated free product (violating FA) or a surjection onto Z (also violating FA). Special cases handle cyclic groups and non-cyclic groups separately. Consequences for groups with Property (T) follow immediately. The paper includes an explicit example for the dihedral group of order 8 and notes AI-assisted methodology.",
"recommendation": "accept",
"required_revisions": [],
"significance": "This settles an open question on the existence of just finite presentations for finite groups. The result is constructive and applies uniformly to all finitely presented groups with Property (FA), including all finite groups. It also yields just-(T) presentations for groups with Kazhdan\u0027s Property (T). The proof technique, based on controlled replacement of relations while preserving the group but destroying fixed-point properties on trees, may have further applications in geometric group theory.",
"strengths": [
"Resolves a named open problem (Kourovka 21.10) with a uniform constructive proof.",
"The just-(FA) strengthening is natural and immediately yields the just-(T) corollary via Watatani\u0027s theorem.",
"Proof is self-contained, uses only standard facts (Serre\u0027s characterization of FA, Neumann\u0027s lemma, and the structure of the relevant semidirect products).",
"Explicit example for D_8 illustrates the construction and verifies irredundancy.",
"Deficiency preservation (Theorem 6) is a useful additional statement."
]
}