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arxiv: 2604.15303 · v2 · submitted 2026-04-16 · 🧮 math.GR

Diameter bounds for arbitrary finite groups and applications

Sean Eberhard , Elena Maini , Luca Sabatini , Gareth Tracey This is my paper

Pith reviewed 2026-05-10 09:30 UTC · model grok-4.3

classification 🧮 math.GR
keywords group diameterCayley graphscomposition factorssoluble groupsGrigorchuk conjecturepermutation groupsBabai conjecture
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The pith

The diameter of a finite group is bounded solely by the diameters of its composition factors and the exponent of its largest normal abelian section.

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

This paper proves a general bound on the diameter of the Cayley graph for any finite group. The bound depends only on the diameters of the group's composition factors and the maximal exponent appearing in a normal abelian section. A sympathetic reader would care because it reduces the study of group diameters to simpler building blocks without requiring full knowledge of the group's structure. This leads to new estimates for soluble groups, permutation groups, and actions on trees. It also advances understanding of long-standing conjectures in group theory under certain conditions.

Core claim

We prove a strong general-purpose bound for the diameter of a finite group depending only on the diameters of its composition factors and the maximal exponent of a normal abelian section. There are a number of notable applications: if G is a finite soluble group of exponent e, diam(G) ≪ e (log |G|)^8; anabelian groups with bounded-rank composition factors have polylogarithmic diameter; transitive soluble subgroups of S_n have diameter ≪ n^5; and Grigorchuk's gap conjecture holds for any finitely generated group acting faithfully on a bounded-degree rooted tree. Additionally, conditional on Babai's conjecture, any transitive permutation group of degree n has diameter bounded by a polynomial n

What carries the argument

A reduction of the group diameter to the diameters of composition factors combined with the maximal exponent of any normal abelian section.

If this is right

  • Finite soluble groups of exponent e satisfy diam(G) ≪ e (log |G|)^8.
  • Anabelian groups with bounded-rank composition factors have polylogarithmic diameter.
  • Transitive soluble subgroups of S_n have diameter ≪ n^5.
  • Grigorchuk's gap conjecture holds for finitely generated groups acting faithfully on bounded-degree rooted trees.
  • Conditional on Babai's conjecture, transitive permutation groups of degree n have polynomial diameter in n, and the gap conjecture holds for residually finite groups.

Where Pith is reading between the lines

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

  • The bound suggests that diameter questions for general groups can be settled once they are settled for simple groups.
  • Explicit versions of the bound could be useful for computational verification of small groups.
  • The approach may extend to give diameter controls for groups with restricted composition factors even when the groups themselves are infinite.
  • It reduces Grigorchuk's conjecture to the case of simple groups for residually finite examples.

Load-bearing premise

The diameters of all composition factors and the maximal exponent of normal abelian sections must be known or bounded independently in advance.

What would settle it

A specific finite soluble group of exponent e whose diameter exceeds any fixed multiple of e (log |G|)^8 would falsify the main application of the bound.

read the original abstract

We prove a strong general-purpose bound for the diameter of a finite group depending only on the diameters of its composition factors and the maximal exponent of a normal abelian section. There are a number of notable applications: (1) if $G$ is a finite soluble group of exponent $e$, $\mathrm{diam}(G) \ll e (\log |G|)^8$, (2) anabelian groups with bounded-rank composition factors have polylogarithmic diameter, (3) transitive soluble subgroups of $S_n$ have diameter $\ll n^5$, and (4) Grigorchuk's gap conjecture holds for any finitely generated group acting faithfully on a bounded-degree rooted tree. Additionally, conditional on Babai's conjecture, (5) any transitive permutation group of degree $n$ has diameter bounded by a polynomial in $n$ (a folkloric conjecture), and (6) Grigorchuk's gap conjecture holds for residually finite groups, and thus the conjecture reduces to the simple case.

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

1 major / 2 minor

Summary. The manuscript proves a general-purpose upper bound on the diameter of an arbitrary finite group G that is expressed in terms of the diameters of the composition factors of G and the maximal exponent of any normal abelian section of G. The bound is then applied to obtain explicit diameter estimates for soluble groups of exponent e (diam(G) ≪ e (log |G|)^8), anabelian groups whose composition factors have bounded rank, transitive soluble subgroups of S_n (diameter ≪ n^5), and to confirm Grigorchuk's gap conjecture for finitely generated groups acting faithfully on bounded-degree rooted trees; conditional on Babai's conjecture the paper also deduces polynomial diameter bounds for all transitive permutation groups of degree n and reduces the gap conjecture to the simple case.

Significance. If the central theorem is correct, the result supplies a flexible reduction tool that converts diameter questions for general finite groups into questions about their composition factors and abelian sections. This yields concrete progress on several open problems, including explicit polylogarithmic and polynomial bounds in the applications listed above and a conditional resolution of Grigorchuk's gap conjecture for residually finite groups. The explicit functional forms given in the applications (including the (log |G|)^8 factor) are a strength, as they make the result immediately usable for further work.

major comments (1)
  1. [Abstract; main theorem (likely Theorem 1.1)] Abstract and statement of the main theorem: the claim that the diameter bound 'depends only on the diameters of its composition factors and the maximal exponent of a normal abelian section' is inconsistent with the functional form derived in application (1). For a soluble group of exponent e the composition factors are cyclic of order at most e and the maximal abelian-section exponent is e, so any function of these two quantities alone is independent of |G|. The appearance of an extra (log |G|)^8 factor therefore implies that the proof accumulates multiplicative contributions across the length of a composition series; the main theorem statement must be revised to make this dependence explicit (e.g., by including the composition length or an equivalent |G|-dependent term).
minor comments (2)
  1. [Abstract] The abstract would benefit from a one-sentence statement of the precise functional form of the main bound rather than the current qualitative description.
  2. [Applications] In the applications section, verify that all implicit constants are tracked consistently when the composition length is inserted into the bound; a short remark on the origin of the exponent 8 would help readers.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying this important point of clarification in the abstract and main theorem. We agree that the dependence on composition length must be made explicit and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract; main theorem (likely Theorem 1.1)] Abstract and statement of the main theorem: the claim that the diameter bound 'depends only on the diameters of its composition factors and the maximal exponent of a normal abelian section' is inconsistent with the functional form derived in application (1). For a soluble group of exponent e the composition factors are cyclic of order at most e and the maximal abelian-section exponent is e, so any function of these two quantities alone is independent of |G|. The appearance of an extra (log |G|)^8 factor therefore implies that the proof accumulates multiplicative contributions across the length of a composition series; the main theorem statement must be revised to make this dependence explicit (e.g., by including the composition length or an equivalent |G|-dependent term).

    Authors: The referee is correct that the current wording of the abstract and Theorem 1.1 is imprecise. The proof of the main bound proceeds by induction along a composition series and accumulates a multiplicative factor depending on the number of steps in the series (which is at most O(log |G|)). This is why the soluble-group application acquires the extra (log |G|)^8 factor even though the composition factors are cyclic of order ≤ e and the maximal normal abelian exponent is e. We will revise the abstract to state that the bound depends on the diameters of the composition factors, the maximal exponent of a normal abelian section, and the length of a composition series. We will likewise update the statement of the main theorem (Theorem 1.1) to make this dependence explicit, for example by including an explicit factor of the composition length or an equivalent |G|-dependent term. These are purely expository changes that do not alter the proofs or the applications. revision: yes

Circularity Check

0 steps flagged

No significant circularity; bound derived from independent external quantities

full rationale

The central theorem asserts a diameter bound expressed in terms of the diameters of composition factors and the maximal exponent of a normal abelian section. These inputs are external to the target diameter and are not defined in terms of it. The applications, such as the (log |G|)^8 factor for soluble groups of exponent e, arise from accumulating contributions over the composition length (which is implicit in the group structure but not a redefinition of the diameter itself). No self-definitional steps, fitted inputs renamed as predictions, load-bearing self-citations, or ansatz smuggling are exhibited by any quoted equation or claim in the abstract or description. The derivation remains self-contained and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The abstract invokes standard facts from finite group theory (existence of composition series, normal abelian sections) but introduces no new free parameters, invented entities, or ad-hoc axioms visible at this level of detail.

axioms (2)
  • standard math Every finite group possesses a composition series whose factors are simple groups.
    Implicit in the statement that the bound depends only on the diameters of the composition factors.
  • standard math Normal abelian sections exist and possess a well-defined maximal exponent.
    Used to control the abelian part of the diameter bound.

pith-pipeline@v0.9.0 · 5475 in / 1348 out tokens · 30161 ms · 2026-05-10T09:30:05.363708+00:00 · methodology

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