Short presentations of finite simple groups
Pith reviewed 2026-05-24 16:15 UTC · model grok-4.3
The pith
Errors in short presentations of alternating and symmetric groups can be fixed while preserving the optimal bit length of O(log n).
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The errors identified in the relevant arguments can be fixed so that the corrected 3-generator 7-relator presentations for A_n and S_n are valid and achieve bit-length O(log n) for n ≥ 5.
What carries the argument
The set of three generators together with the seven relations that define the alternating and symmetric groups, after the supplied corrections.
If this is right
- The presentations satisfy all the given relations for each n at least 5.
- The total bit length remains proportional to the logarithm of n.
- This length is the smallest possible because specifying n requires at least that many bits.
- The presentations can be written down explicitly once the corrections are applied.
Where Pith is reading between the lines
- This method of locating and correcting errors in group presentations might apply to constructions for other families of groups.
- Valid short presentations could simplify computational checks of group properties for large n.
- Further work might seek presentations with even fewer relations or generators for the same groups.
Load-bearing premise
The supplied fixes are enough to ensure that the generators satisfy every one of the seven relations at the same time.
What would settle it
Direct substitution of the proposed generators into each of the seven relations to check whether each relation evaluates to the identity element for some specific n at least 5.
read the original abstract
Guralnick, Kantor, Kassabov and Lubotzky (J. Eur. Math. Soc. 13.2, 2011, 391-458) [GKKL] give 3-generator 7-relator presentations of $A_n$ and $S_n$ with bit-length $O(\log n)$ for $n\geq5$. This is the best possible bit-length, since $\Omega(\log n)$ bits are required to specify the integer $n$ in the input. However, the generators do not satisfy the relations. This paper considers the relevant arguments given in [GKKL], identifies where the errors occur, and shows how they can be fixed in order to recover this result. The presentations are available in Magma.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper identifies specific errors in the 3-generator 7-relator presentations of A_n and S_n claimed by Guralnick-Kantor-Kassabov-Lubotzky (GKKL, JEMS 2011), supplies explicit corrections to the generators and relations, and asserts that the corrected presentations are valid with bit-length O(log n) for n ≥ 5. Magma code implementing the presentations is provided for verification.
Significance. If the corrections are shown to work, the result restores the best-possible bit-length bound for presentations of these groups and supplies concrete, machine-checkable data that can be used in computational group theory and in the study of short presentations of finite simple groups.
major comments (1)
- [Abstract and the section describing the fixes] The central claim that the GKKL result is recovered rests on the corrected generators satisfying all seven relations simultaneously for every n ≥ 5. The abstract states that errors were located and fixed, but the manuscript must supply either a uniform algebraic verification or a complete computational check (via the supplied Magma code) that no additional relations fail after the listed corrections; without this, the homomorphism from the presented group to A_n/S_n is not guaranteed.
Simulated Author's Rebuttal
We thank the referee for the careful reading and the constructive major comment. We address it directly below.
read point-by-point responses
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Referee: [Abstract and the section describing the fixes] The central claim that the GKKL result is recovered rests on the corrected generators satisfying all seven relations simultaneously for every n ≥ 5. The abstract states that errors were located and fixed, but the manuscript must supply either a uniform algebraic verification or a complete computational check (via the supplied Magma code) that no additional relations fail after the listed corrections; without this, the homomorphism from the presented group to A_n/S_n is not guaranteed.
Authors: We agree that explicit confirmation is required. The Magma code accompanying the paper constructs the corrected generators parametrically in n and directly tests whether they satisfy all seven relations in A_n and in S_n. In the revised manuscript we will add a short subsection (in the section describing the fixes) that (i) states that the supplied code performs this check, (ii) reports that the relations hold for all tested n from 5 to several hundred, and (iii) notes that the uniform algebraic form of the corrections ensures the same verification applies for every larger n. This supplies the required computational evidence for the surjective homomorphism. revision: yes
Circularity Check
No circularity: explicit correction of external reference with independent verification
full rationale
The paper identifies concrete errors in the external GKKL reference and supplies explicit corrected generators and relations for A_n/S_n, supported by publicly available Magma code for external checking. No step reduces by definition, by fitted parameter renamed as prediction, or by self-citation chain; the central claim rests on direct comparison to prior work plus machine verification outside the paper's own fitted values or ansatzes.
Axiom & Free-Parameter Ledger
axioms (2)
- standard math The axioms of group theory hold for the symmetric and alternating groups under consideration.
- domain assumption Magma correctly evaluates whether given elements satisfy given relations in a finitely presented group.
discussion (0)
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