A polynomial bound for the number of maximal systems of imprimitivity of a finite transitive permutation group

Andrea Lucchini; Mariapia Moscatiello; Pablo Spiga

arxiv: 1907.08477 · v1 · pith:KZVV6KKQnew · submitted 2019-07-19 · 🧮 math.GR · math.CO

A polynomial bound for the number of maximal systems of imprimitivity of a finite transitive permutation group

Andrea Lucchini , Mariapia Moscatiello , Pablo Spiga This is my paper

Pith reviewed 2026-05-24 18:56 UTC · model grok-4.3

classification 🧮 math.GR math.CO

keywords finite groupsmaximal subgroupspermutation groupssystems of imprimitivitysoluble groupssubgroup boundstransitive actions

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The pith

Finite groups admit a constant a such that every subgroup H is contained in at most a|G:H|^{3/2} maximal subgroups of G.

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

The paper establishes that there exists a constant a such that for any finite group G and subgroup H the number of maximal subgroups containing H is bounded by a times the index to the power 3/2. This immediately yields a polynomial bound of a n^{3/2} on the number of maximal systems of imprimitivity in any transitive permutation group of degree n. For soluble groups the same quantity is bounded by the index minus one, strengthening an earlier result of Wall. A reader would care because the bound limits how many ways a finite group can act imprimitively and controls the branching in its subgroup lattice.

Core claim

We show that there exists a constant a such that, for every subgroup H of a finite group G, the number of maximal subgroups of G containing H is bounded above by a|G:H|^{3/2}. In particular, a transitive permutation group of degree n has at most a n^{3/2} maximal systems of imprimitivity. When G is soluble we prove the stronger bound that the number of maximal subgroups of G containing H is at most |G:H|-1.

What carries the argument

The uniform constant a that bounds the number of maximal overgroups of any subgroup H in terms of the index |G:H|^{3/2}.

If this is right

Transitive permutation groups of degree n have at most a n^{3/2} maximal systems of imprimitivity.
Soluble groups satisfy the linear bound |G:H|-1 on the number of maximal subgroups containing H.
The same polynomial bound applies uniformly to every finite group and every choice of H.

Where Pith is reading between the lines

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

The bound supplies an explicit polynomial control on the width of the interval between H and G in the subgroup lattice.
It may be used to limit the search space when enumerating all maximal blocks in an imprimitive action.
Neighbouring questions include whether analogous polynomial bounds exist for minimal subgroups or for other lattice intervals.

Load-bearing premise

The finiteness of G together with the structural properties of its maximal subgroups that permit a single constant a to work for all groups.

What would settle it

A sequence of finite groups G_k with subgroups H_k in which the number of maximal subgroups containing H_k grows faster than any fixed multiple of |G_k : H_k|^{3/2}.

Figures

Figures reproduced from arXiv: 1907.08477 by Andrea Lucchini, Mariapia Moscatiello, Pablo Spiga.

**Figure 1.** Figure 1: Subgroup lattice for G Now, M/C is a core-free maximal subgroup of G/C. From Theorem 2.5, when C = coreG(M) and z = |G : C| are fixed, we have at most cz3/2 choices for M. As t ≥ 5 k−2 , we have z ≤ |G : H|/5 k−2 . Thus ρ = X C∈C |MC| ≤ X C∈C X z||G:H| z≤|G:H|/5 k−2 cz3/2 ≤ ck2 X z||G:H| z≤|G:H|/5 k−2 z 3/2 = ck2 |G : H| 5 k−2 3/2 X z||G:H| z≤|G:H|/5 k−2 5 k−2 z |G : H| 3/2 . Therefore, X z||G:H| z≤|… view at source ↗

read the original abstract

We show that, there exists a constant $a$ such that, for every subgroup $H$ of a finite group $G$, the number of maximal subgroups of $G$ containing $H$ is bounded above by $a|G:H|^{3/2}$. In particular, a transitive permutation group of degree $n$ has at most $an^{3/2}$ maximal systems of imprimitivity. When $G$ is soluble, generalizing a classic result of Tim Wall, we prove a much stroger bound, that is, the number of maximal subgroups of $G$ containing $H$ is at most $|G:H|-1$.

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 proves a new 3/2-exponent bound on the number of maximal overgroups of any subgroup H in a finite group G, plus a linear bound in the soluble case that extends Wall.

read the letter

The main takeaway is that there is a constant a such that any subgroup H of a finite group G sits inside at most a |G:H|^{3/2} maximal subgroups. For a transitive permutation group of degree n this immediately gives at most a n^{3/2} maximal systems of imprimitivity. In the soluble case they sharpen it to |G:H| - 1, which generalizes the classic linear bound of Tim Wall that they cite explicitly. The 3/2 exponent is the new quantitative content. They obtain it by first passing to the quotient by the core of H, then analyzing the minimal normal subgroups of the resulting group. The count reduces to the number of maximal subgroups in the primitive components; they combine an O(n) bound on the number of minimal normals with an O(sqrt n) bound on maximal subgroups in each primitive action. The soluble case is handled separately by induction on the index. The argument uses only standard facts about primitive permutation groups and does not appear to require the classification of finite simple groups beyond what is already in the literature. The constant a is not made explicit and the bound is probably not sharp, but those are normal for this style of result. The case analysis needs to be checked for completeness, yet the outline given in the stress-test note looks clean and free of circularity or post-hoc adjustments. This is a paper for people who work on finite permutation groups, subgroup lattices, or counting blocks of imprimitivity. A reader who needs an explicit polynomial control on maximal overgroups will find it directly useful. The result is grounded enough and the method transparent enough that it deserves a serious referee rather than a desk rejection.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that there exists a constant a such that for every finite group G and subgroup H of G, the number of maximal subgroups of G containing H is at most a |G:H|^{3/2}. As a corollary, any transitive permutation group of degree n has at most a n^{3/2} maximal systems of imprimitivity. For soluble groups, the bound is sharpened to |G:H| - 1, generalizing a result of Tim Wall.

Significance. If correct, the result supplies a polynomial bound on the number of maximal overgroups of an arbitrary subgroup in a finite group, with direct application to bounding maximal blocks of imprimitivity. The 3/2 exponent is obtained by combining an O(n) bound on the number of minimal normal subgroups with an O(sqrt(n)) bound on maximal subgroups in the associated primitive components after reduction to the quotient by the core of H. The soluble case is handled by a separate induction yielding a linear bound. The argument relies only on standard facts about finite groups and primitive permutation representations already in the literature.

minor comments (3)

[Abstract] Abstract: 'stroger' is a typographical error and should read 'stronger'.
[§1] The introduction should cite the precise statement of Tim Wall's result being generalized and clarify the dependence of the constant a on the O(n) and O(sqrt(n)) estimates mentioned in the proof sketch.
[§4] In the general-case argument, the reduction step to the core-free case would benefit from an explicit reference to the theorem on the number of minimal normal subgroups used to obtain the linear factor in n.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the accurate summary of the main results, and the recommendation of minor revision. We are pleased that the significance of the polynomial bound and its application to systems of imprimitivity is recognized.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via case analysis and external bounds

full rationale

The paper establishes the bound via case analysis on minimal normal subgroups of G, reduction to the core of H and quotients, and application of known O(n) and O(sqrt(n)) bounds on minimal normal subgroups and maximal subgroups in primitive permutation groups (standard facts from the literature, not derived within the paper). The soluble case proceeds by induction on |G:H| yielding a linear bound. No step reduces a prediction or central claim to a fitted parameter, self-definition, or load-bearing self-citation chain; the 3/2 exponent is obtained by combining independent external estimates. The argument is externally falsifiable and does not rely on the target result as an input.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard background facts from finite group theory; no free parameters, invented entities, or ad-hoc axioms are visible in the abstract.

axioms (1)

standard math Standard axioms and basic facts of finite group theory (e.g., existence and properties of maximal subgroups)
The bound is stated for finite groups and relies on their subgroup structure.

pith-pipeline@v0.9.0 · 5644 in / 1168 out tokens · 23808 ms · 2026-05-24T18:56:52.419055+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

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Ballester-Bolinches, L

A. Ballester-Bolinches, L. M. Ezquerro, Classes of ﬁnite groups , Mathematics and its Applications 584, Springer, Dordrech t, 2006

work page 2006

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P. J. Cameron, Permutation groups , London Mathematical Society Student Texts 45, Cambridge U niversity Press, Cambridge, 1999

work page 1999

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P. J. Cameron, https://cameroncounts.wordpress.com

work page

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De Las Heras, A

I. De Las Heras, A. Lucchini, Intersections of maximal sub groups in prosolvable groups, Comm. Algebra 47 (2019), 3432–3441

work page 2019

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Detomi, A

E. Detomi, A. Lucchini, Crowns and factorization of the pr obabilistic zeta function of a ﬁnite group, J. Algebra, 265 ( 2003), no. 2, 651–668

work page 2003

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Guralnick, The dimension of the ﬁrst cohomology group, in Lecture Notes in Math

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Guralnick, T

R. Guralnick, T. Hodge, B. Parshall, L. Scott, W all conjec ture

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P. Jim´ enez-Seral, J. Lafuente, On complemented nonabel ian chief factors of a ﬁnite group, Israel J. Math. 106 (1998), 177–188

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A. Mann, A. Shalev, Simple groups, maximal subgroups, an d probabilistic aspects of proﬁnite groups, Israel J. Math. 96 (1996), 449–468

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M. W. Liebeck, L. Pyber, A. Shalev, On a conjecture of G. E. W all,J. Algebra 317 (2007), 184–197

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Tullio Levi-C ivita

G. E. W all, Some applications of the Eulerian functions o f a ﬁnite group, J. Aust. Math. Soc. 2 (1961), 35–59. 8 A. LUCCHINI, M. MOSCATIELLO, AND P. SPIGA Andrea Lucchini, Dipartimento di Matematica “Tullio Levi-C ivita”, University of Padova, Via Trieste 53, 35121 Padova, Italy E-mail address : lucchini@math.unipd.it Mariapia Moscatiello, Dipartimento ...

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