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arxiv: 2606.26030 · v1 · pith:HAGCLFJ7new · submitted 2026-06-24 · 🧮 math.GR

Computing canonical labellings of finite solvable groups

Santiago Barrera Acevedo , Heiko Dietrich , Max Horn This is my paper

Pith reviewed 2026-06-25 19:22 UTC · model grok-4.3

classification 🧮 math.GR
keywords canonical labellingfinite solvable groupsgroup isomorphismgroup presentationsisomorphism testingcomputational group theory
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The pith

Finite solvable groups get canonical presentations that match exactly when the groups are isomorphic.

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

The paper defines a canonical labelling function that assigns to every finite solvable group a specific presentation describing an isomorphic copy. Two such groups are isomorphic if and only if they receive identical presentations, and the construction supplies an explicit isomorphism from the original group to the labelled one. This turns isomorphism testing into a direct comparison of labels and supplies a concrete map between groups. The method adapts presentation techniques from p-groups together with cohomology and automorphism algorithms to cover all finite solvable groups and includes a working implementation sketch.

Core claim

We define a canonical labelling function on the class of finite solvable groups so that two such groups G and H are isomorphic if and only if can(G)=can(H). Specifically, can(G) is a group presentation that describes a group isomorphic to G, and our description explains how to construct an isomorphism G to can(G). The approach combines canonical presentation methods for p-groups with cohomology and automorphism group algorithms.

What carries the argument

The canonical labelling function can(G) that produces a unique group presentation together with an explicit isomorphism to the original group.

If this is right

  • Isomorphism of two finite solvable groups reduces to equality of their computed presentations.
  • An explicit isomorphism from any given finite solvable group to its canonical form can be constructed.
  • All finite solvable groups become comparable by a uniform labelling that supports storage and retrieval.
  • The labelling extends the scope of existing presentation methods from p-groups to the full solvable class.

Where Pith is reading between the lines

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

  • The labels could support building searchable collections of solvable groups ordered by their presentations.
  • Refinements might reduce computation time for groups of moderate order where current steps are slowest.
  • Similar structural decompositions in other group classes could admit parallel labelling constructions.

Load-bearing premise

Techniques for canonical presentations of p-groups can be extended using cohomology and automorphism computations to cover every finite solvable group without obstruction.

What would settle it

Two non-isomorphic finite solvable groups that receive identical canonical presentations, or a concrete finite solvable group on which the labelling procedure cannot be completed.

Figures

Figures reproduced from arXiv: 2606.26030 by Heiko Dietrich, Max Horn, Santiago Barrera Acevedo.

Figure 1
Figure 1. Figure 1: Example computation for two small groups gap> all := AllSmallGroups(44100,IsSolvableGroup);; Size(all); 3087 # there are 3087 solvable groups of size 44100 = 2ˆ2.3ˆ2.5ˆ2.7ˆ2 gap> ### create two identical lists of copies of these 3087 groups gap> grps := List(all, x->PcGroupCode(RandomSpecialPcgsCoded(x),Size(x)));; gap> grps2:= List(grps,x->PcGroupCode(CodePcGroup(x),Size(x)));; gap> List(grps, CanonicalPc… view at source ↗
Figure 2
Figure 2. Figure 2: Example computation for solvable groups of size 44100 References [1] S. Barrera Acevedo, H. Dietrich, M. Horn. The GAP package CanonicalPcPres. gap-packages.github.io/CanonicalPcPres (to appear soon, 2026). [2] H. U. Besche, B. Eick. Construction of finite groups. J. Symb. Comput. 27:387-404 (1999), DOI: 10.1006/jsco.1998.0258. [3] H. U. Besche, B. Eick, E. A. O’Brien. A millennium project: constructing sm… view at source ↗
Figure 3
Figure 3. Figure 3: Example computations for three larger groups with de￾rived lengths 2, 4, and 5, respectively [6] J. J. Cannon, B. Eick, C. R. Leedham-Green, Charles. Special polycyclic generating sequences for finite soluble groups. J. Symb. Comput. 38:1445-1460 (2004), DOI: 10.1016/j.jsc.2004.05.003. [7] J. J. Cannon, D. F. Holt. Automorphism group computation and isomorphism testing in finite groups. J. Symb. Comput. 35… view at source ↗
read the original abstract

We define a canonical labelling function on the class of finite solvable groups so that two such groups $G$ and $H$ are isomorphic if and only if can$(G)=$can$(H)$. Specifically, can$(G)$ is a group presentation that describes a group isomorphic to $G$, and our description explains how to construct an isomorphism $G\to$can$(G)$. Our approach is motivated by O'Brien's (1993) canonical presentations for finite $p$-groups and utilises ideas from group cohomology first described by Robinson (1982) and automorphism group algorithms developed by Smith (1994), Holt (2001), and others. We also discuss a proof-of-concept implementation for the computer algebra system GAP and comment on the major bottlenecks and open research questions.

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.

Referee Report

2 major / 2 minor

Summary. The paper defines a canonical labelling function can(G) on finite solvable groups such that G ≅ H iff can(G) = can(H), where can(G) is an explicit group presentation together with a constructive isomorphism G → can(G). The construction proceeds by building a composition series, using Robinson cohomology to solve extension problems and produce canonical presentations at each step, and applying existing automorphism-group algorithms (Smith/Holt style) to resolve choices; it is motivated by O'Brien's 1993 canonical presentations for p-groups and includes a proof-of-concept GAP implementation together with discussion of bottlenecks and open questions.

Significance. If the construction is correct and fully algorithmic, the result would supply a practical canonical form for the important class of finite solvable groups, extending O'Brien-style methods beyond p-groups and enabling direct isomorphism testing by string comparison of presentations. The explicit use of cohomology and aut-group routines, together with the GAP implementation, constitutes a concrete algorithmic contribution that could be adopted in computational group theory libraries.

major comments (2)
  1. [motivation and approach paragraph] The central construction (composition series + cohomological extension solving + aut-group resolution) is described at a high level in the approach paragraph; a load-bearing gap is the lack of an explicit algorithm or pseudocode showing how the Robinson cohomology cocycle data is converted into a canonical presentation when the extension is non-split (see the paragraph on motivation and approach).
  2. [discussion of open research questions] The claim that the method 'extends without obstruction' to all finite solvable groups relies on the assumption that the chosen composition series and cohomology computations always terminate with a unique canonical representative; no termination or uniqueness argument is supplied for groups whose composition factors have non-trivial outer automorphisms (see the discussion of open research questions).
minor comments (2)
  1. [abstract] The abstract cites Robinson (1982), Smith (1994) and Holt (2001) but does not indicate which specific results (e.g., which cohomology theorem or aut-group algorithm) are invoked; adding one-sentence pointers would improve readability.
  2. [GAP implementation paragraph] The GAP implementation is described only as 'proof-of-concept'; including at least one concrete example (e.g., a small non-p-group solvable group together with its computed can(G)) would make the practicality claim easier to assess.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the identification of areas where additional detail would strengthen the manuscript. We address the two major comments below.

read point-by-point responses

  1. Referee: [motivation and approach paragraph] The central construction (composition series + cohomological extension solving + aut-group resolution) is described at a high level in the approach paragraph; a load-bearing gap is the lack of an explicit algorithm or pseudocode showing how the Robinson cohomology cocycle data is converted into a canonical presentation when the extension is non-split (see the paragraph on motivation and approach).

    Authors: We agree that the current description of the non-split case remains at a high level. In the revised manuscript we will insert explicit pseudocode (approximately one page) that spells out the conversion from Robinson cocycle data to the canonical presentation, including the steps that invoke the automorphism-group routines to resolve choices. revision: yes

  2. Referee: [discussion of open research questions] The claim that the method 'extends without obstruction' to all finite solvable groups relies on the assumption that the chosen composition series and cohomology computations always terminate with a unique canonical representative; no termination or uniqueness argument is supplied for groups whose composition factors have non-trivial outer automorphisms (see the discussion of open research questions).

    Authors: The manuscript already frames the extension to arbitrary solvable groups as subject to open questions precisely when outer automorphism groups are non-trivial. We do not assert an unconditional termination or uniqueness proof; the text instead lists the relevant obstructions as topics for further work. We will revise the wording in the open-questions section to remove any phrasing that could be read as claiming an obstruction-free extension. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines an explicit algorithmic construction for a canonical labelling can(G) on finite solvable groups by combining composition series, Robinson cohomology for extensions, and existing automorphism-group algorithms (O'Brien, Smith, Holt). This is a constructive definition rather than a derivation that reduces to its inputs. No equations, fitted parameters, or predictions appear; cited works are independent prior results with no author overlap or load-bearing self-citation chains. The central claim is self-contained as a definition plus construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies insufficient technical detail to enumerate free parameters, axioms, or invented entities; all fields left empty.

pith-pipeline@v0.9.1-grok · 5655 in / 1009 out tokens · 20848 ms · 2026-06-25T19:22:53.241863+00:00 · methodology

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Reference graph

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