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

Recognition: unknown

Diameter bounds for arbitrary finite groups and applications

Elena Maini, Gareth Tracey, Luca Sabatini, Sean Eberhard

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

classification 🧮 math.GR

keywords group diameterCayley graphscomposition factorssoluble groupsGrigorchuk conjecturepermutation groupsBabai conjecture

0 comments

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.

Desk Editor's Note private letter to a colleague

The paper reduces group diameters to composition-factor data plus abelian exponents and derives some concrete applications, but the abstract overstates how independent the bound is from |G|.

read the letter

The main takeaway is a general bound on the diameter of any finite group G that is controlled by the diameters of its composition factors together with the largest exponent appearing in a normal abelian section. From that the authors extract bounds for soluble groups, anabelian groups, transitive soluble subgroups of S_n, and some conditional statements about Babai's conjecture and Grigorchuk's gap conjecture for tree actions and residually finite groups. That reduction itself is the genuinely new piece; earlier work handled special cases but did not give a uniform way to pass from the simple factors and the abelian data to the whole group. The applications to soluble groups and to bounded-rank composition factors are worked out cleanly and look like direct consequences. The conditional results are presented as such, which keeps the claims honest. The soft spot is exactly the one the stress-test raises. The abstract says the bound depends only on the composition-factor diameters and the maximal abelian exponent, yet the soluble-group corollary carries an extra (log |G|)^8 factor. Any function of those two quantities alone cannot produce |G|-dependent terms, so the proof must be accumulating contributions across the composition series; the abstract should have stated the dependence on series length or on |G| explicitly. The rest of the paper appears to be careful, but that mismatch between claim and application is worth a referee's attention. This is for people who work on diameters of finite groups or on the two conjectures mentioned. A reader who already knows the literature on Babai and Grigorchuk will see the value immediately. The work shows clear thinking and honest engagement with prior results, so it deserves a serious referee. I would send it out for review.

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)

[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)

[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.
[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

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

discussion (0)

Reference graph

Works this paper leans on

58 extracted references · 3 canonical work pages

[1]

Aschbacher,On the maximal subgroups of the finite classical groups, Invent

M. Aschbacher,On the maximal subgroups of the finite classical groups, Invent. Math. 76(1984), no. 3, 469–514.↑10

1984
[2]

Babai, P

L. Babai, P. J. Cameron, and P. P. P´ alfy,On the orders of primitive groups with restricted nonabelian composition factors, J. Algebra79(1982), no. 1, 161–168.↑5

1982
[3]

Babai and ´A

L. Babai and ´A. Seress,On the diameter of permutation groups, European J. Combin. 13(1992), no. 4, 231–243.↑1, 7, 18

1992
[4]

Bajorska and O

B. Bajorska and O. Macedo´ nska,A note on groups of intermediate growth, Comm. Algebra35(2007), no. 12, 4112–4115.↑6

2007
[5]

Bajpai, D

J. Bajpai, D. Dona, and H. A. Helfgott,Growth estimates and diameter bounds for untwisted classical groups, arXiv e-prints (2021), available at https://arxiv.org/ abs/2110.02942.↑2

work page arXiv 2021
[6]

Bartholdi,The growth of Grigorchuk’s torsion group, Internat

L. Bartholdi,The growth of Grigorchuk’s torsion group, Internat. Math. Res. Notices 20(1998), 1049–1054.↑30

1998
[7]

,Lower bounds on the growth of a group acting on the binary rooted tree, Internat. J. Algebra Comput.11(2001), no. 1, 73–88.↑31

2001
[8]

Bartholdi and A

L. Bartholdi and A. Erschler,Growth of permutational extensions, Invent. Math.189 (2012), no. 2, 431–455.↑6

2012
[9]

Bartholdi, R

L. Bartholdi, R. I. Grigorchuk, and Z. ˇSuni´k,Branch groups, Handbook of algebra, Vol. 3, 2003, pp. 989–1112.↑7, 30, 31, 32, 33

2003
[10]

Black,Asymptotic growth of finite groups, J

S. Black,Asymptotic growth of finite groups, J. Algebra209(1998), no. 2, 402–426. ↑33

1998
[11]

Breuillard, B

E. Breuillard, B. Green, and T. Tao,Approximate subgroups of linear groups, Geom. Funct. Anal.21(2011), no. 4, 774–819.↑2

2011
[12]

Breuillard and M

E. Breuillard and M. C. H. Tointon,Nilprogressions and groups with moderate growth, Adv. Math.289(2016), 1008–1055.↑29 34 SEAN EBERHARD, ELENA MAINI, LUCA SABATINI, AND GARETH TRACEY

2016
[13]

de la Harpe,Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000.↑30, 31

P. de la Harpe,Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000.↑30, 31

2000
[14]

J. D. Dixon and B. Mortimer,Permutation groups, Graduate Texts in Mathematics, vol. 163, Springer-Verlag, New York, 1996.↑22

1996
[15]

Dona,The diameter of products of finite simple groups, Ars Math

D. Dona,The diameter of products of finite simple groups, Ars Math. Contemp.22 (2022), no. 4, Paper No. 6, 8.↑18, 19

2022
[16]

J. R. Driscoll and M. L. Furst,On the diameter of permutation groups, Proceedings of the fifteenth annual acm symposium on theory of computing (stoc ’83), 1983, pp. 152–160.↑1

1983
[17]

Eberhard and E

S. Eberhard and E. Maini,The growth of residually soluble groups, arXiv e-prints (2025), available athttps://arxiv.org/abs/2511.07018.↑6, 8, 23, 26, 29

work page arXiv 2025
[18]

Eberhard and L

S. Eberhard and L. Sabatini,Expanding groups with large diameter, arXiv e-prints (2026), available athttps://arxiv.org/abs/2602.13582.↑29

work page arXiv 2026
[19]

Erschler,Not residually finite groups of intermediate growth, commensurability and non-geometricity, J

A. Erschler,Not residually finite groups of intermediate growth, commensurability and non-geometricity, J. Algebra272(2004), no. 1, 154–172.↑6

2004
[20]

Erschler and T

A. Erschler and T. Zheng,Growth of periodic Grigorchuk groups, Invent. Math.219 (2020), no. 3, 1069–1155.↑31

2020
[21]

M. D. Fried,Introduction to modular towers: generalizing dihedral group–modular curve connections, Recent developments in the inverse Galois problem (Seattle, WA, 1993), 1995, pp. 111–171.↑30

1993
[22]

M. D. Fried and M. Jarden,Field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 11, Springer, Cham, 2023. Fourth edition of 868860, Revised by Moshe Jarden.↑30

2023
[23]

S. P. Glasby,The composition and derived lengths of a soluble group, J. Algebra120 (1989), no. 2, 406–413.↑5, 9

1989
[24]

R. I. Grigorchuk,On the Hilbert-Poincar´ e series of graded algebras that are associated with groups, Mat. Sb.180(1989), no. 2, 207–225, 304.↑25, 26

1989
[25]

,On the gap conjecture concerning group growth, Bull. Math. Sci.4(2014), no. 1, 113–128.↑6

2014
[26]

In preparation.↑6

,A note on gaps and just-infiniteness, 2026. In preparation.↑6

2026
[27]

Gromov,Groups of polynomial growth and expanding maps, Inst

M. Gromov,Groups of polynomial growth and expanding maps, Inst. Hautes ´Etudes Sci. Publ. Math.53(1981), 53–73.↑6

1981
[28]

R. M. Guralnick,Cyclic quotients of transitive groups, 2000, pp. 507–532. Special issue in honor of Helmut Wielandt.↑21

2000
[29]

R. M. Guralnick, A. Mar´ oti, and L. Pyber,Normalizers of primitive permutation groups, Adv. Math.310(2017), 1017–1063.↑5

2017
[30]

Halasi, A

Z. Halasi, A. Mar´ oti, L. Pyber, and Y. Qiao,An improved diameter bound for finite simple groups of Lie type, Bull. Lond. Math. Soc.51(2019), no. 4, 645–657.↑2

2019
[31]

H. A. Helfgott,Growth in linear algebraic groups and permutation groups: towards a unified perspective, Groups St Andrews 2017 in Birmingham, 2019, pp. 300–345.↑18

2017
[32]

H. A. Helfgott and ´A. Seress,On the diameter of permutation groups, Ann. of Math. (2)179(2014), no. 2, 611–658.↑2

2014
[33]

Kassabov and I

M. Kassabov and I. Pak,Groups of oscillating intermediate growth, Ann. of Math. (2) 177(2013), no. 3, 1113–1145.↑33

2013
[34]

E. I. Khukhro and V. D. Mazurov,Unsolved Problems in Group Theory. The Kourovka Notebook (v41), arXiv e-prints (2026), available at https://arxiv.org/abs/1401. 0300.↑33

2026
[35]

Kleidman and M

P. Kleidman and M. Liebeck,The subgroup structure of the finite classical groups, London Mathematical Society Lecture Note Series, vol. 129, Cambridge University Press, Cambridge, 1990.↑12, 13, 22

1990
[36]

Kohl,A bound on the order of the outer automorphism group of a finite simple group of given order(2003), available at https://stefan-kohl.github.io/preprints/ outbound.pdf

S. Kohl,A bound on the order of the outer automorphism group of a finite simple group of given order(2003), available at https://stefan-kohl.github.io/preprints/ outbound.pdf. Preprint.↑17 DIAMETER BOUNDS FOR ARBITRARY FINITE GROUPS 35

2003
[37]

Kornhauser, G

D. Kornhauser, G. L. Miller, and P. G. Spirakis,Coordinating pebble motion on graphs, the diameter of permutation groups, and applications, SIAM Journal on Computing 13(1984), no. 2, 330–344.↑1

1984
[38]

L. G. Kov´ acs and C. E. Praeger,On minimal faithful permutation representations of finite groups, Bull. Austral. Math. Soc.62(2000), no. 2, 311–317.↑21

2000
[39]

Landazuri and G

V. Landazuri and G. M. Seitz,On the minimal degrees of projective representations of the finite Chevalley groups, J. Algebra32(1974), 418–443.↑14

1974
[40]

Lev,On large subgroups of finite groups, J

A. Lev,On large subgroups of finite groups, J. Algebra152(1992), no. 2, 434–438. ↑12

1992
[41]

M. W. Liebeck and A. Shalev,Simple groups, permutation groups, and probability, J. Amer. Math. Soc.12(1999), no. 2, 497–520.↑5

1999
[42]

Lubotzky and A

A. Lubotzky and A. Mann,On groups of polynomial subgroup growth, Invent. Math. 104(1991), no. 3, 521–533.↑26

1991
[43]

Mann,How groups grow, London Mathematical Society Lecture Note Series, vol

A. Mann,How groups grow, London Mathematical Society Lecture Note Series, vol. 395, Cambridge University Press, Cambridge, 2012.↑30

2012
[44]

Mar´ oti,On the orders of primitive groups, J

A. Mar´ oti,On the orders of primitive groups, J. Algebra258(2002), no. 2, 631–640. ↑28

2002
[45]

McKenzie,Permutations of bounded degree generate groups of polynomial diameter, Inform

P. McKenzie,Permutations of bounded degree generate groups of polynomial diameter, Inform. Process. Lett.19(1984), no. 5, 253–254.↑1

1984
[46]

N. E. Menezes,Random generation and chief length of finite groups, ProQuest LLC, Ann Arbor, MI, 2013. Thesis (Ph.D.)–University of St. Andrews (United Kingdom). ↑17

2013
[47]

Milnor,Growth of finitely generated solvable groups, J

J. Milnor,Growth of finitely generated solvable groups, J. Differential Geometry2 (1968), 447–449.↑8

1968
[48]

Nekrashevych,Palindromic subshifts and simple periodic groups of intermediate growth, Ann

V. Nekrashevych,Palindromic subshifts and simple periodic groups of intermediate growth, Ann. of Math. (2)187(2018), no. 3, 667–719.↑6

2018
[49]

M. F. Newman,The soluble length of soluble linear groups, Math. Z.126(1972), 59–70.↑23

1972
[50]

P. P. P´ alfy,A polynomial bound for the orders of primitive solvable groups, J. Algebra 77(1982), no. 1, 127–137.↑5

1982
[51]

Pyber,Asymptotic results for permutation groups, Groups and computation (New Brunswick, NJ, 1991), 1993, pp

L. Pyber,Asymptotic results for permutation groups, Groups and computation (New Brunswick, NJ, 1991), 1993, pp. 197–219.↑5

1991
[52]

Pyber and E

L. Pyber and E. Szab´ o,Growth in finite simple groups of Lie type, J. Amer. Math. Soc.29(2016), no. 1, 95–146.↑2

2016
[53]

D. A. Suprunenko,Soluble and nilpotent linear groups, American Mathematical Society, Providence, RI, 1963.↑10

1963
[54]

45, American Mathematical Society, Providence, RI, 1976

,Matrix groups, Translations of Mathematical Monographs, Vol. 45, American Mathematical Society, Providence, RI, 1976. Translated from the Russian, Translation edited by K. A. Hirsch.↑10

1976
[55]

Tracey,Invariable generation of permutation and linear groups, J

G. Tracey,Invariable generation of permutation and linear groups, J. Algebra524 (2019), 250–289.↑10

2019
[56]

J. S. Wilson,On the growth of residually soluble groups, J. London Math. Soc. (2)71 (2005), no. 1, 121–132.↑5, 9, 26

2005
[57]

,The gap in the growth of residually soluble groups, Bull. Lond. Math. Soc.43 (2011), no. 3, 576–582.↑5, 7, 15, 26

2011
[58]

T. R. Wolf,Solvable and nilpotent subgroups of GL(n, q m), Canadian J. Math.34 (1982), no. 5, 1097–1111.↑5 Mathematics Institute, Zeeman Building, University of Warwick, UK Email address:firstname.lastname@warwick.ac.uk

1982