A Database of Groups with Equivalent Character Tables
Pith reviewed 2026-05-24 19:49 UTC · model grok-4.3
The pith
A database partitions all finite groups of order less than 2000 into classes with identical character tables.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The authors have built a database containing, for all finite groups of order less than 2000 excluding those of order 1024, a partitioning of the groups into classes where two groups lie in the same class precisely when a permutation of the rows and a permutation of the columns of one character table produces the other.
What carries the argument
Conversion of each character table into a graph, followed by computation of a canonical form via graph-isomorphism software and hashing of the resulting graph.
If this is right
- Any two groups placed in the same class share every numerical invariant that can be read from a character table.
- The database supplies an immediate answer to whether any two listed groups have equivalent tables.
- The method scales to orders containing hundreds of millions of groups by combining graph-canonicalization with high-throughput distributed computation.
- Orders such as 1536 receive complete partitions that were previously unavailable.
Where Pith is reading between the lines
- The same graph-hashing technique could be applied to groups of larger order once sufficient computational resources become available.
- The database supplies a ready source of examples for testing conjectures about which group invariants are determined by the character table.
- Patterns visible in the partition sizes may suggest which orders tend to produce many groups with identical tables.
Load-bearing premise
The graph-isomorphism and hashing procedure correctly identifies every pair of tables that are equivalent under row and column permutations and never groups non-equivalent tables together.
What would settle it
Discovery of two groups of order less than 2000 whose character tables become identical after row and column permutation yet are placed in different classes by the database.
read the original abstract
Two groups are said to have the same character table if a permutation of the rows and a permutation of the columns of one table produces the other table. The problem of determining when two groups have the same character table is computationally intriguing. We have constructed a database containing for all finite groups of order less than 2000 (excluding those of order 1024), a partitioning of groups into classes having the same character table. To handle the 408,641,062 groups of order 1536 and other orders with a large number of groups we utilized high-throughput computing together with a new algorithmic approach to the problem. Our approach involved using graph isomorphism software to construct canoncial graphs that correspond to the character table of a group and then hashing the graphs.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper constructs a database that partitions all finite groups of order less than 2000 (excluding order 1024) into equivalence classes of groups having identical character tables. Equivalence is defined via row and column permutations. For orders with many groups (e.g., 408641062 groups of order 1536), the authors convert each character table to a graph, compute a canonical form using graph-isomorphism software, and hash the result to identify matching tables; high-throughput computing is used to scale the enumeration.
Significance. If the computational method is shown to be correct, the resulting database would be a substantial reference resource for the study of character-table equivalence in finite groups, potentially enabling new investigations into the relationship between group isomorphism and character-table isomorphism. The scale of the computation (particularly for order 1536) and the use of graph canonization for this algebraic problem would also constitute a methodological contribution.
major comments (3)
- [Abstract] Abstract and methods: the encoding that converts a character table (with complex algebraic entries) into a graph is not described. It is therefore impossible to verify that distinct entry values receive distinct vertex colors or auxiliary structures, that the bipartite row-column incidence is augmented to record exact values rather than equality patterns only, and that the encoding is applied uniformly across all groups.
- [Abstract] Abstract: no validation is reported against known small-order cases (e.g., groups of order < 100) where character-table equivalences have already been classified in the literature or in the GAP/SmallGroups library. Without such checks or sample outputs, the correctness of the partitions cannot be assessed.
- [Abstract] Abstract (order-1536 case): the manuscript supplies neither an error-rate estimate nor an independent verification procedure for the 408641062 groups of order 1536. Because the central claim is the correctness of the resulting partition, the absence of any soundness argument for the largest computational component is load-bearing.
minor comments (1)
- [Abstract] The abstract refers to “canoncial graphs”; this is presumably a typographical error for “canonical graphs.”
Simulated Author's Rebuttal
We thank the referee for the constructive comments highlighting areas where additional detail and verification would strengthen the manuscript. We address each major comment below and commit to revisions that improve verifiability without altering the core computational results.
read point-by-point responses
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Referee: [Abstract] Abstract and methods: the encoding that converts a character table (with complex algebraic entries) into a graph is not described. It is therefore impossible to verify that distinct entry values receive distinct vertex colors or auxiliary structures, that the bipartite row-column incidence is augmented to record exact values rather than equality patterns only, and that the encoding is applied uniformly across all groups.
Authors: We agree the encoding procedure requires explicit description for reproducibility. The manuscript provides only a high-level statement that canonical graphs are constructed from character tables. In the revised version we will insert a dedicated Methods subsection that specifies: (i) the mapping of distinct algebraic values (including complex numbers) to vertex colors or labels, (ii) the precise augmentation of the bipartite row-column graph to encode exact values rather than mere equality patterns, and (iii) confirmation that the same encoding is applied uniformly to every group. This addition will make the conversion process fully verifiable. revision: yes
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Referee: [Abstract] Abstract: no validation is reported against known small-order cases (e.g., groups of order < 100) where character-table equivalences have already been classified in the literature or in the GAP/SmallGroups library. Without such checks or sample outputs, the correctness of the partitions cannot be assessed.
Authors: We will add an explicit validation subsection. It will report direct comparisons of our partitions against published character-table equivalence classifications and the GAP SmallGroups library for all groups of order less than 100, together with sample outputs for representative cases. These checks will be presented to demonstrate consistency with existing results before scaling to larger orders. revision: yes
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Referee: [Abstract] Abstract (order-1536 case): the manuscript supplies neither an error-rate estimate nor an independent verification procedure for the 408641062 groups of order 1536. Because the central claim is the correctness of the resulting partition, the absence of any soundness argument for the largest computational component is load-bearing.
Authors: We acknowledge that an explicit soundness argument for the order-1536 computation is necessary. The original manuscript relies on the determinism of standard graph-canonization libraries and collision-resistant hashing but does not quantify error rates or describe independent checks at this scale. In revision we will add a dedicated paragraph discussing computational reliability, including any subset cross-checks performed, the known correctness properties of the canonization software, and a clear statement of remaining limitations. Because exhaustive independent verification of 408 million tables is computationally prohibitive, the revision will be partial on this point. revision: partial
Circularity Check
No circularity: direct computational enumeration using external tools
full rationale
The paper performs an exhaustive computational classification of finite groups by order, converting each character table to a graph, applying external graph-isomorphism software to obtain canonical forms, and hashing the results to partition groups. No quantity is defined in terms of itself, no parameter is fitted to a subset and then re-used as a prediction, and no uniqueness theorem or ansatz is imported via self-citation to justify the central partitioning. The method is self-contained against external benchmarks (graph-isomorphism libraries) and produces the claimed database by explicit enumeration rather than any deductive chain that reduces to its inputs.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption Two groups have the same character table if a permutation of the rows and a permutation of the columns of one table produces the other table.
discussion (0)
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